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Orifl 

begi 

the 

sion 

oth( 

first 

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The 
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TINI 
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Map 
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begi 
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Lorsque  le  document  est  trop  grand  pour  dtre 
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et  de  haut  en  bas,  en  prenant  le  nombre 
d'images  ndcessaire.  Les  diagramnes  suivants 
illustrent  la  m^thode. 


1 

2 

3 

1 

2 

3 

4 

5 

6 

THE  LIBRARY 


F  HE  UNIVERSITY  OF 
BRITISH  COLUMBIA 


J 

] 


INVESTIGATION 


OP 


CORRECTIONS  TO  HANSEN'S  TABLES  OF  THE  MOON; 


•vT:a   n 


•4 


TAHLKS   von  TIIKIU   vV  rPLICATlON. 


BV 


S  T  IVt  O  N    iNl  E  W  C  < )  M  H  , 


I'UoKKssoi;,  r.  s.  n.vvv. 


FORMING  PART  III  OF  PAPERS  PURUSIIED  BY  THE  COMMISSION  ON  THE  TRANSIT  OF  VENUS. 


WASHINGTON: 

a  O  V  E  R  N  M  E  N  r     P  11  I  N  T  I  N  a     O  F  I'M  C 

1870. 


diiltBCHHSHBaesSS 


TABLE    OF    CONTENTS 


'^^l 


Vagi'. 


INTROIILCTORV  NOTE 


i 


i  i.-INVICSncJATION  or  ERRORS  OF  LONGITUDE. 

Evection 8 

Variation 8 

Mraii  urior  iif  (alxiiar  liylit  a>cciisii)n  at  <liiri'iLnt  tiiius  of  (lay i) 

N'aliic  of  solar  paiallax  cniploved i) 

l.isi  of  corrci  lions  to  "  .\ri;iimtiit  Fonilaiiuiilal  ' lo 

(HMK'ral  Ideas  which  form  tlic  liasis  of  this  investigation li 

Dill'ercntial  cofllicients 12 

Mean  apparent  error  of  Hansen's  Tallies  in  riyht  asiension 12 

Siiilden  apparent  alteration  in  mean  motion  of  the  moon 1  1 

Corrections  for  limli  and  oliservatory  lo  render  oljservalioK;.  sirielly  eoniparalile 13 

Mean  ontstaiulinn  talinlar  error  of  the  moon  in  lon.nitiule 13 

Corrections  of  sliort  perioil  acliially  applied 14 


Ivpiation  connectini;  eriurs  of  moon's  laludar  right  ascension  with  errors  of  Innar  elements 

Hnins  ol  eriors  of  moon's  1  orr^ried  right  ascensitiii  given  In  ohservalioiis  at  (ireenwich  .ind  Washington.      ...  17 

Normal  ei|ualions  foi  deteriiiiTiing  (1/,  «,  and  /(■  liy  least  sipiares 20 

Values  of  outstanding  errois  of  lunar  elements  for  each  year 20 

.Apparent  periodic  character  of  the  corrections  10  the  eccentricity  and  perigee 20 

Formuhc  for  the  new  inequality  of  longitude 24 

Discussion  of  tircenwich  observations  of  the  moon  Iroin  1S47  to  1S5S 24 

Sums  of  residual  errors 2(1 

Corrections  to  eccentricity,  longitude  of  perigee,  and  iiioon's  longitude 21) 


S  2— 1N\'ESTI(;.\TI()N  OF  P()L.\R   DlSl'.VNCF,  .\NI)  I,.\TrrUDE. 

Corrections  to  dei  lination  ilepending  on  errors  of  loiigiiude 

Constant  corrections  lo  reduce  declinations  to  same  luiHlaiiieiilal  standard 

Sums  of  errors  of  moon's  corrected  declination,  given  liy  oliservaiions  at  (ireenwich  and  Washington. 
Correction  lo  inclination  of  orliit  and  longilude  of  node 


32 
32 
34 
36 


^  3.— Al'XILI.\RV    T.\UI.i:S    FOR    FACILITA'riNt;     ITIK   COMPUTATION    OF    TDK   CORRECTIONS 

HANSEN'S  "TAHLES  DE  I,A   LUNE  '. 


TO 


Summarv  of  corrections  to  mean  and  true  longitude  of  the  moon  from  Hansen's  Tables 37 

E.\planation  of  tables  for  applying  these  corrections 37 

Example  of  the  use  of  the  tables 411 

Corrections  to  the  Ephemeris  ilerived  from  Uaiisen's  Tables  of  the  Moon,  for  Greenwich  mean  noon  of  each  dav, 

from  1S74.  September  I,  to  1875,  January  31 41 

Tables  I,  H,  HI.  the  arguments,-, /),../,  A',   11 45 

Tables  IV,  V,  VI,  secular  and  empirical  terms 4f) 

Table  VH,  lernis  of  luean  longitude 47 

I'able  VIII,  terms  of  true  longitude ^      .  48 

Tables  IX,  X,  factors  lor  reduction  lo  longitude  in  orbit ;  and  for  correction  of  latitude  and  reduction  to  ecliptic 

longitude 50 

Table  XI,  factors  lor  convertin,g  small  changes  of  longilude  and  latitude  into  changes  of  right  ascension  and  decli- 
nation   1:1 


.tk%'^'w^    '.:  ^ '"'I 


6 


INTRODUCTORY    NOTE. 


6 


Wiicii  llie  proItliMii  (iC  iililizin",'  llie  oltservalioiisorocciiUiitidiis  at  (he  sovcnil  Transit, 
of  V<;iHis  siatioiis,  so  as  to  (Ictcriniuc  tin;  loiigiliidcs  of  tliosc;  .slalions  willj  all  atlaiiial»l<! 
acciiiacy,  was  presented  to  the  Comniission  on  the  Transit  of  Venus,  it  was  found  neces- 
sai-}'  to  make  a  careful  detcrniinatiou  of  the  errors  of  the  lunar  ephenieris  liefore  an 
entirely  satisfactory  solution  of  the  prohleui  couhl  he  atten^pted.  The  Secretary  of  (Ik; 
C'oinniission  was  therefore  charged  with  this  work,  most  of  the  computations  on  which 
have  been  made  und<'r  his  direction  l»y  Mr.  1).  1?.  Todd,  computer  for  the  Commission. 

Wasiuxoton,  M<nj  25,  1876. 


I  f 


mu-^. 


CO  kR  i:CTI()NS    '!■()    15  !■    A  I' I' I.  I  !•  I )    TO   1 1  A  N  S  !•:  N  '  S     I'AIU.I'.S 

Ol"     111  !•:    MOON. 


m 


iN\Ksri(;.\ii(i\  ()!•■  EkK">Ks  (»|-  i,f)\(;i  riDK. 

One  (if  llic  mosl  iin|H»i-taiil  npcijiliuiis  in  cniiiicclidii  willi  llic  oltsciviilioiis  ol  llir 
Iraiisil  of  Venus  is  fin-  acciiriilc  (Iclcniiiiiiilioii  (»!'  llic  loiiniliidcs  nl  llic  sir.lioiis.  Mmiiv 
ot"  these  slaliniis  arc  so  liir  leiiioved  IVdiii  lclci,napliic  eoiiiiiiiiiiiciifioii  llial  I  lie  loiiiriliMles 
must  <lc|»eii<l  iiiaiiilv  on  llic  moon.  Dclerniinalions  of  loii^nliiile  iVoiii  iihkiii  enlniina- 
tinns  an;  lonnd  Ity  e\|ierieiiec  lo  lie  snhjerl  lo  e(Mislaiil  eirurs  w  liieli  il  is  dillieiill  lo 
(lelcrinine  and  allow  for.  Il  was  llienjlore  a  pari  of  llie  policy  ol"  Ihc  Aiiieri<'an  Coiii- 
niission  lo  depeii<l  on  occnilalioiis  rallier  lliaii  iip(ni  moon  enlminalioiis  litr  lli"dclermi- 
na1i(»n  ol  l(Mi<riliides.  The  reason  Tor  lliis  course  is,  1  lial  llic  disappearaiwe  ol' a  slar 
hohind  Ihe  linih  <d'  tin'  moon  is  a  sudden  phcnonicnon,  llie  lime  ol'  wliieli  can  alwavs  he 
lix<!d  williin  a  fnielion  of  :i  second.  If  llic  eplicmeris  ol"  the  moon  and  s!ar  were  cor- 
recl,  and  ilit^  trisk  oi  iiic  (uriiM  i  ,i  peii'i!cl  ciirie,  llic  lon^nlndi'  could  We  dclciniincd 
from  Ihe  oc(Millalion  wilh  Ww.  saiiu;  dctirce  of  acnirac}'  llial.  the  pheiMniienon  <'oidd  he 
ohserved.  'J'lie  <picslion  arises,  how  rarlhcse  sources  ol"  error  can  he  diminished.  The 
iiie(|iialitics  of  llu^  lunar  surface  liniii  a  source  of  error  w  liich  il  is  iinpossilile  lo  avoid, 
hut  which  is  comparalivcly  innocuous  when  many  oh.servalions  arc  made,  since  Ihe 
errors  will  he  purely  accideiilal,  and  will  llicrcfor(^  he  eliminalr'd  from  llic  mean  ol  a 
great  number  of  ol)sorva(ioiis. 

Tlio  position  of  llu'  slar  ran  ho  dclermiiied  hy  meridian  ohservalions  wilh  alinosl 
any  rerpiired  degree  of  accniiu'y.  We  have,  llieii.  only  lo  see  how  liir  Ihe  errors  of  Ihe 
lunar  c|)hem('ri.s  can  he  diminished:  :iiid  lo  reduce  lliese  errors  lo  a  ininimaiii  is  the 
ohject  ol   the  present  paper. 

Hansen's  lahles  are  tak<Mi  for  this  purpose,  lieeaiisc  I  here  is  reason  lo  heliev(!  that 
the  porta rhat ions  on  which  these  tahles  are  ioiinded  are,  in  the  main,  exlreinelv 
accurate;  more  accurate  and  C(niipl»(le,  in  faci,  llian  any  others  which  liiivc  heen 
tabulated.  Still,  before  they  can  be  used  lln'the  purpose  in  <pieslion,  a  number  oi' verv 
important  corrections  arc  rcrjuin'd,  which  wi^  may  divide  inio  two  classes, — {'(U'lcclioiis 
to  the  theory,  and  lo  the  elements. 

It  is  well  known  that  Ilanscm  increased  all  tln^  iterturbations  of  his  tables  by  the 
consliuit  factor  0.0001544,  on   account  of  a  supi)osed  want  cd'  coiiicidencf!  between  the 


m 


tf'iilcr  ul'  lliriin;  iiiul  (lu!  ccnttT  oC  ji^nivify  of  llic  iiioon.     I  liiivc  sliowii  tliii(    iriiitscii 

fails  to  Niisliiiii  tills  |)usifi(iii,  and  that  tlii'ir  is  no  ^rtind  reason  to  snpjtosc  that  tlic  moon 

(liirt'rs  iVoni  any  other  of  tin;   heavenly  bodies  in  this  respeet.*     Oiir  lirst  course  would 

(liorefori!  Ite  (o  diminish  all  of  Hansen's  ineiiualities  by  this  liietor,  \ver(>  il  not  that,  tlnro 

are  reasons  why  each  of  the  two  greatest,  pertnrltations  of  the  moon's  motion, — the  evec- 

tionand  tin;  variation, — shonid  Ix;  found  lar;;er  from  <d)servation  than  he  fonnd  them  iVom 

theory. 

ErvctUm. — Tlio  evection  has  the  eccentricity  as  allictor;   the  value  of  the  otht-r 

liictor  being  nearly  0.4.     if,  then,  the  ad(»|»ted  ccciMitricity  of  tin;  motm   be  erroneous, 

the  computed  evection  will  be  erronc(nis  by  four-tenths  the  anninnt  of  the  erntr.      Now, 

by  reference  to  Hansen's   ^^ Jhii/ri^inii:;  t/ir  thioirlischcn  licircliiniiii;  ilir  in  ihii  Minulhi- 

ffin  aiigiintii(tl('ii  Stiinini;c)i"\  (pa,<fe  173),  it  will  In;  seen  that  the  eccentricity  adopted 

thronjfhout  in   the  compidation  of  the  pertnrbatiims  of  tin;  moon  is  less  by  o.ooocxi7^ 

than  (he  vahu!    he   linally  fonnd   from   observation,  and  adopted   in  the  tables,      iiiid  lie 

nsed  tim  latter  valne,  the   Ihecn'elical   evretion  would    have  been  i^reuler   by  Ibe  liactinu 

.000007^  ,,,,      ,.    ,  ,11  1  I    .  .  ,         ,  I        • 

rro.oooi^V       llie  lac(or  actualU  used  liriiiif ').(.)( K)t  s  4),  tlic  t'\('<'li(iii,  I  liiis  lu- 
.0549C)()S 

creased,  is  t(»o   lar<(e   by  only  (J.00002  i    ol   its  entire  amount,  or  o".o().     ( 'ousc(|Meutly, 

the  tabular  coeltieient  of  evection  should  be  diminished  by  this  amount.      Precisely  the 

same  result  follows,  if  we  adopt   Hansen's  view  of  a   separatiim  ot'  the  eenti'rs  of  liy^ure 

and  irravity  of  lh<-  moon:  and  llauseii  himself  is  led  to  il  on  paiie  1/5  of  the  work  cited, 

only  instead  ot'  o".og,  he  says,  "kein  vidles  Zelmlheil  <'iner  Secuutle.'' 

Vdiitilion. — That  th(!  coellicicut  o!  varialinu  residtinjj;  from  meridian  obseivatious 
will  be  jrreaterthan  the  actual  coelilcienl  may  be  anticipated  from  the  litHowinLM-ou- 
siderations.  The  iueipudity  iu  ipu-slion  attains  its  maxima  and  minima  in  the  moon's 
octants  In  the  lirst  octant,  wr  have  a  nniximnm.  The  ehmgalion  of  the  nu)on  from 
the  sun  is  then  alxMit  ,^'';  ami  the  observed  position  ot  the  union  is  mainly  dependent  on 
observations  of  the  first  limb  made  in  tlnMlaytime, wlntn  the  apparent  semi-dianicter  of 
the  moon  will  b<;  diminished  by  the  brilliancy  of  the  surroundiiii;  sky.  No  account  of 
this  diminntjoii  of  the  iij.parent  semi-tliameler  beiii<;  taken  in  tin;  reductions,  the  semi- 
diameter  actually  applied  is  too  lari^e,  and  tin;  (diserved  right  ascension  of  the  moon  is 
al.so  too  larg(\ 

When  th.  moon  reacdies  the  third  octant,  the  valne  of  the  variation  attains  its  inin- 
imiim.  The  moon  then  transits  at  9'',  and  the  meridian  observation  is  made  (Mi  the  lirsl 
limb,  while  the  apparent  s«Mui-diametcr  is  increased  i»y  tlu;  i:radiatioii  coiiseipuMit  upon 
the  contrast  betwecMi  tlu;  moon  and  (Im  sky.  The  result  will  b(!  that  the  observed  right 
ascension  will  be  too  small. 

The  same  causes  will  mak(!  the  observed  right  asoMisioii  too  great  in  the  fifth 
octant,  and  too  small  in  the  sciventh.  'i'hese  positive  and  negative  errors  of  ol)served 
right  ascension  correspond  to  the  limes  of  maximum  and  minimum  ell'ects  of  variation 
ill  iiicriMising  the  longitude  of  the;  moon.     Thend'ore,  the  observed  variation  will  appa- 

*  J'rociMMliiijfH  of  till)  Aiiiitricitri  Associiitiuu  for  tliu  AdvnucoiURDt  of  yuiencn,  i86S, — Hilliiiiaii'H  Anieriuati  Journal 
of  Seuiiicn,  Noveiiilier,  iS6S. 

t  Altliaiiillniigvii  tk>r  iiiiiUii'matisvh-iiliyHiMclien  CliiSHe  dor  Krmi);Ucli-8iivhHiNc1icii  GusellHclmft  der  WiMoiiHulinften 
Baud  vl. 


0 

rciitly  1»(!  Iiiru[<'i'  tliiiii  llic  iicliial  vniijitiitii.  wlialcvcr  lliis  iniiy  lie.  'I'liis  secno  a  niiirli 
more  natural  and  [triiltaMc  cause  tor  tlic  a|)|)an<nt  excess  of  tlw  ohscrvcil  over  tlic  theoreti- 
cal perturlmlions  tlian  tlial  assigned  l»y  Hansen.  Hansen's  factor  onlv  increases  tin-  coetli- 
cient  in  (jin'stion  l>yo".3;,;  Init  it  seems  prolialile  lliat  llie  variation  derived  from  obser- 
vations alone  woidil  he  yet  lai^'er  than  Hansen's  increased  variation.  In  tiict,  in  iSi);,  I 
tlinnd,  hy  coni|tarin<,f  the  errors  of  the  Innar  epliemeris  when  th<!  moon  cidminated  at 
dill'erent  times  (d'  the  day,  that  the  eireet  of  llie  <frrater  irradiation  at  niyht  was  very 
8tr(Hii,dy  nuirked.  Dnrinii  the  linir  years  1X62-65  tlie  mean  I'rrors  of  the  iaiiles  in 
right  ascension  at  diilerent  tinn's  of  day  werr  as  follows:* 

I. 

Heforo  snnsd —  o.  1 54 

A  Her  briifht.  daylis,'ht  in  the  eveniiii;' — 0.093 

l>el(ire  liriijht  ilayliij;ht  in  tli     morniiiif- .  .      -fo.ogi 
After  sunrise -|- O'  '  5.1 

In  the  dilli'rence  hctween  the  resnits  lor  e;i' h  limli,  the  ellect  of  increa.sed  irradia- 
tion seems  to  he  o".o6. 

The  only  icmaininu:  term  which  is  larire  enoniili  to  lie  materiall.v  all'eeted  liy  the" 
increasf!  in  (|neslion  is  ihe  annual  equation,  ol'  wlii(di  the  increaM    is  o".io. 

A  ii[lance  at  the  errors  (d'  Hansen's  taldes,  ^fiven  liy  nn-ridian  oliservalions,  will  show 
that,  the  errors  ahont  the  time  of  lirst  (|narter,  and,  indeed,  dnriuij;  the  first  half  of  the 
lunation,  are  in  the  nu'an  h'ss  l»y  helween  3"  and  4"  than  dm-injj;  Ihe  seeoml  half. 
Ilence,  either  the  semi-dianmter,  in-  tlw!  parallactic  e(|natioii,  or  liolli,  an^  loo  larye.  The 
parallactic  etpnition  nse<l  hy  Hansen  citrres|)onds  to  a  value  .S".9i6  for  the  solar  paral- 
lax, which  value  is  too  larir(>  hy  prohaldy  not  much  less  than  o".io.  The  result 
which  I  deduced  in  US67  from  all  tin;  really  valuahle  data  exiani  was  .S",S4,S  ;  and  Ihe 
determinations  \vhi(h  have  since  been  made,  when  revised  with  the  h.«^t  data,  seem  to 
indicate  a  diniinnti(Mi  of  this  value  rather  than  an  increase.  These  indications  are,  how- 
ever, a.s  yet,  ii  little  loo  indelinil((  to  predicate  ariythinif  upon.  I  shall  I heretore  con- 
tinue; to  n.se  S".84S,  which  will  dindnish  Hansen's  value  hy  o".o6S.  The;  correspond! njf 
diminution  in  the!  ]trin<'ipal  parallactic  term  will  he  o".()6,  while  there  will  he  two  other 
terms  to  receive  a  smaller  dimiiiulion. 

This  correction  will  still  leave  a  diH'erence  (d"  ahout  ::"  helween  i\u\  results  from 
the  first  and  second  limbs,  which  will  be  accounted  for  by  an  error  of  1"  in  theado]»ted 
semi-diameter.  This  correction  to  the  semi-diameler  is  a  priori  tpnte  probable,  as  Ihe 
improved  meridian  instruments  of  the  present  lime  give  a  .send-diameter  of  the  sun  1" 
less  than  Ihe  older  ones  from  wliieli  the  diameters  adopted  in  onr  ej)hemerides  were 
derived.  It  is  to  Ik;  expected  that  Ihe  .stni' diumeler  of  the  moon  will  exhibit  a  sim- 
ilar apparent  diminution. 

From  a  note  in  I  lansen's  Ihtrleginis!;  (|>age  439),  it  w  ill  be  seen  that  one  of  the  terms 
in  the  true  longitude  has  cre|d  into  the  tables  with  a  wrong  sign.  AscMnployed  in  lhetai)les, 
and  given  on  page  15  of  the  introduction,  it  is,  -f  o".335  sin  (25-  — 4  "•'  +  2f.)— 40'). 

As  revised  in  Wwlhtrlegnng,  it  is —  o".285  sin 

Theretbre  the  tables  need  the  correction —  o".62    sin 

*  Iuveiitig.-ktiou  of  the  Distance  of  tlio  Sun,  p.  24. 
2  M 


'I 


w 


Kl 


? 


^4' 


10 

The  f«»llo\vii)g  is  alist  of  llic  roriTctinns  wo  Imvo  so  far  deduced  to  Hansen's  tables. 
Tliev  should  in  slri(;lness  l)e  applied  to  Ihe  mean  longllude,  or  '^Argiin/pn/  fondu/ncndir, 
■l»nt  they  may  without  serious  error  !)(>  applied  to  IIk;  true;  lonyiliide. 
Put 

J>,  the  argiunent  ol'  parallactie  ineqnalily,  or  mean  elongalion  of  the  moon  from 

the  sun  ; 
iT,    the  moim's  mean  anomaly : 
g',  the  sun's  mean  anomaly  ; 

fo,   the  disfanee  ol"  the  moon's  perigee;  from  the;  aseending  node; 
&>',  tiie  distanee  of  (he  sun's  perigee  from  the  same  node. 


We  then  have 

and  the  correetions  in  (piestion  ar 

■  0.96  sii 
0.07  sin  (P 


J)  —   n  _  g'  -(_  f.)  . 


M 


4-0.96  sin    7'  ^ 

—  IT    )       ■  rnvnUnetn-  U'niis. 


—  O.I ;,  sin  (/>+-')    > 

-{-0.09  sin  ii'  Anniinli'iiinlH.n. 
0.33  sin    2    /)  Viniall.;!. 

—  o.  10  sin  (3  D  —  ir)     f:,',rii„ii. 

—  0.63  sin  {  ^  2  —  4  i,''  -|-  2  r.>  —  4  m' 


Ai'riilental  error. 


The  fourth  and  filih  terms  of  this  expression  lia\e  the  elfeel  to  remove  the  iiierease 
wliieli  Hansen  applied  to  his  inetjualities  on  aeeonnt  ol  the  jtosition  of  the  eenter  of 
gravity  of  the  moim,  while  the  sixth  is  the  residt  of  the  slight  error  of  the  eeeentrieity 
wliieh  lie  emidoyeii  in  eompiiting  the  coetVici(!nt  of  eveetion. 

In  comparing  with  nn'riilian  (d)servafions  which  have  l»een  reduced  without  any 
correction  to  the  apparent  semi-dianiet(,'r  depending  on  the  time  of  day,  the  e(»rreelion 
of  variation  may  also  ht;  omitted,  since  a  yet  larger  apparent  correction,  having  the  oppo- 
site alircltraie  siiru.  will  r<,'siilf,  from  the  apparent  variations  of  that  semi-diameter,  as 
ulreaily  explained. 

As  regards  the  possiUh;  correct  ions  to  the  elements  of  Hansen's  tables,  it  is  t(»  he 
renuirked  tlnit  that  investigator  did  not  avail  himself  of  the  elements  of  tin;  lunar  orhit 
deducefl  l>y  Airy  from  thcM  Jreenwich  ohservations  between  1750  mikI  1S30,  but  <dttaiiied 
liis  final  values  of  the  elements  by  a  comparison  of  his  own.  ()!'  the  nature  and  t>xtent 
of  the  observations  thus  employe<l,  we  have  no  details ;  but  it  is  not  likely  that  more 
than  a  very  small  fraction  of  flw'  entire  mass  of  ol)servations  was  used,  and  iti  can  then;- 
fore  hardly  bi'  expected  that  the  elenients  were  det('rmined  with  the  last  degree  of 
accuracy.  Any  error  in  the  motion  of  the  perii;ee  01  node  will  constantly  increase  with 
the  time.  It',  in  addition  to  this,  we  rellect  that  Ihe  meridian  obs<!rvations  of  the  lasj 
twenty  yejirs  are  lin*  more  accurate  than  llios<!  Hansen  had  at  his  dispctsal,  it  will  not 
seem  at  all  surprising  to  liml  (piite  sensible  enors  in  Ihe  present  longitudes  of  tlu!  lunar 
perigee  and  node  as  derived  !»y  Hansen.     Our  lu'xt  step  will  therefore  be  to  d<;termine 


■Wi'; 


11 


■i 
m 


wliat  corrections  to  liuiiscirs  clciiiciils  arc  iiidiciiti'd  l>y  tlir  rccfiit  oltscrvatioiis  ol  tlic 
iiiooii  made  at  (Jrcciiwicli  and  Wasliiii^'loii  .since  iSOJ,  a  period  duiinu  wiiicli  liolli 
series  of  oliservations  aid  carefnllv  conipan-d  witli   llansenV  taldi's. 

Tlie  general  ideas  on  wiiicli  liie  present  invcslii,'atiiin  ot  liie>e,  correeticni.s  is  liased 
are  these:  tile'  errors  ol'  liie  moon's  laiuilar  l(Hii,nliide  are  ol'  Iwo  classes, — a  progressive 
correction,  wliicli  ap|)arently  increases  nniHtrmiy  \\  ilii  tiie  lime;  and  errors  ol  short 
period,  tin'  principal  ones  of  which  go  llironLrh  their  |teriu,l  dnriiii,'  one  revolution  of  the 
moon  (»r  less.  In  determininij;  the  errors  (if  the  (irst  class  I'roni  oh.servation,  those  ol 
IIh^  second  class  may  he  reij;arded  as  accidental  ernns,  the  eli'ect  of  which  will  l>e  elim- 
inaled  from  the  mean  of  a  larire  nnml>er  of  oliservations.  Since,  in  a  .series  ol  ohserva- 
lioiis  e.\l('ndini(  (hronifh  a  mnnher  of  years,  tin;  maxima  and  mininiaof  each  term  ol 
short  period  will  tiill  indiscriminati-ly  into  all  parts  of  all  the  other  [M-riods,  each  periodic 
c()rre(;li(ni  may  he  determined  as  if  the  c[]\'r]s  of  the  others  were  |»nrely  accidental 
errors.  At  the  same  time,  as  the  elimiiialion  of  eaeh  periodic  ernn-  from  tin;  ma.xima 
and  minima  of  all  thi;  (»th(M-s  cannot  he  complete  in  any  iinite  tinn',  it  is  desirahle  that 
each  periodic  cm-rcction  of  sensilde  maiinitnde  which  we  can  determine  beforehand  shall 
he  applied  to  the  rcsi<hials  hcliirc  the  laltcrarc  used  to  determine  thi'  corrections  toth»! 

elements. 

The  corrections  of  the  elcnu-iits  of  loimiliidc  have  been  made  to  depend  principally 
tii)oii  the  observed  riulit  ascensions,  in>tead  ol  rednciny  the  oliserved  ernns  ol  riiihl 
ascensi(ni  and  polar  distance  to  errors  nf  hni-iliide  and  lalitnde.  Thi-  reason  lor  this 
course  is,  that  the  apparent  errors  of  pcdar  distance,  alter  correcting  them  approxinnilely 
for  errors  of  the  elements  easily  (h'terininetl.  will  aii>e  principally  from  ernn,*  of  obhcr- 
vation,  and  not  from  errors  of  I  he  taldes.  In  tact,  I  lie  niisei  vat  ions  of  the  moon's  declina- 
tion are  sometinn's  aU'ected  with  accidental  errors  »['  ii  niaunitmle  which  it  is  ditVicidt  to 
account  f(»r,  especially  in  the  case  of  Washinylon.  (Irantinjr  that  the  moon  moves  in  a 
plane  the  position  of  which  can  be  very  accurately  determined,  we  have  at\erwurd  only 
to  determine  the  moon's  |)()sition  in  that  plane,  and  this  ciin  In- (hnie  from  an  ob.scrved 
right  ascension  almo.st  as  well  as  if  we  had  a  directly  observed  loni,ntnde.  The  longi- 
tude thus  determined  will  be  less  likely  lobe  aU'ecled  with  systematic  errors  than  il  we 
suppose!  the  position  entirely  unknown,  and  chaiii:e  the  einns  of  right  ascension  ami 
declinat'on  to  errors  of  lon^^ilude  and  latitude,  without  regard  to  the  po.ssihle  constant 
errors  of  the  ineasure<l  declinations. 

Foriiiula-  for  expressing  the  longitude  and  latitude  of  the  nuton  in  terms  of  the 
lunar  elements  are  given  by  Hansen  in  a  posthumous  memoir.*  The  following  terms 
are  sullicient  for  our  pres<;nt  jiurpose  : 

Tut 

/,         the  mooifs  hnigiiude  in  (uiiit  : 

0,       the  hmgitude  of  the  ascending  node  : 

/,        the  inclination  of  the  orbit  to  the  ecliptic; 

(t,^,  the  moon's  right  ascensimi  and  declinalioii  ; 

cj,      the  obrnpiity  of  the  ecliplie.  __^__ 


•  II.)b«n-  ili(!  Diirstolluiii;  <lrr  Kiaili'ii  AiirstciKiiiiK  inid  Alnv.ic  lnin-  drs  .MoiuleH  in  Kmictioii  der  LiiiiKo  in  dor  Itnlni 
und  d.T  Knotcniiint;!'.     Ai.!iiiii(nniii;..ii  d.i  KiinislicIi-S.'i.liMsili.u  (J.stlls.liiifl  del-  WlNHt-nHcliuncn,  ltd.  x,  No.  viii. 


12 


■i^^^^ 


Wo  tlieii  have,  apin-oximalcly, 

«_/_2'^.5Hii2/-  i".!  sin(2/-0)+  I ''.I  sill  0 
Mill  5  zz  isiii  ft)  sill  /  +  cos  ay  sin  i  sin  {1  —  0) 
—  0.40  sill  /  +  0.08  sin  {1—0) 
Tiie  (lifleiciilial  co-etHcionts  derived  iVoin  tlicse  cxinvssioiis  are, 
(la 
dl 
da 
do 
da 


—  1  _  0.037  eos  {■?-  I  —  0)  —  0.087  cos  2  / 

—  0.018  cos  61+0018  COS  {2I  —  O) 


-'*-  -  0.2 1  sill  0  —  0.2 1  sin  {2I—O) 
dl 

cos  (5  '^'^  =  0.40  cos  /  +  0.08  COS  (/  —  0) 
dl 


—  (^0.40  +  o.oS  COS  0)  cos  /  +  0.08  sill  0  sin  / 

cos«5  '^'^  =-0.081  co>i{l-0) 
do 

cos  S  "-:   —  0.92  sill  {I  —0) 
dt 

From  the  first  three  forniulie,  it,  will  bo  seen,  that  the  mean  error  in  ri<rht  ascension 
is  very  nearly  the  same  as  the-  mean  error  in  loiiiritnde;  the  i.eriodic  corrections  lieing 
siiitposed  to  he  eliminated  from  this  mean. 

The  investigation  of  the  corrections  fnmi  ohservations  is  now  made  as  luHows : 
All  the  apparent  errors  of  the  tables  derived  from  the  meridian  observations  at  Green- 
vviehaiidWashingtmi  since  1S62  have  been  collected,  arranged  in  the  order  of  dates 
and  the  mean  taken  for  each  year;  observations  of  the  separate  limbs  being  kept  sepa- 
rate. The  mean  error  in  right  ascension  for  each  year  is  as  follows: 
Apparent  cirors  of  Hansen's  tahles  in  Jl.  A. 


Greenwich. 

Diir. 

W.ishington 

Mean. 

Year. 

I. 

II. 

I. 

II. 

Dim 

I. 

11. 

Mean. 

1862 

" 

" 

It 

" 

" 

" 

-  3-(' 

-  0.6 

—  2.1 

l8()3 

. 

. 

-  2.3 

+  0.5 

--  0.9 

1S64 

. 

. 

.      . 

—     I.O 

+    1.3 

+  0.4 

1865 

—  0.2 

+   30 

3.2 

+  0.3 

+  3-9 

3.f> 

0.0 

-1-   3-4 

+   1.7 

1 806 

+    1.2 

+   3-6 

2.4 

+  0.9 

+  4-5 

3.6 

+   1.0 

+   4'J 

+  2.5 

1867 

+  2.4 

+   5-7 

3-3 

+  2.4 

+   5.3 

:.4 

+  2.4 

+   5.8 

+   4-1 

1 808 

<♦-   2.f) 

-1-  0.0 

3-4 

+  2.4 

+  6.6 

4.2 

+   2.5 

-1-   6.3 

+  4-4 

l86() 

^  3-3 

1-   5.6 

2.3 

+  1 

+   7-4 

4.0 

+  3.4 

•I-  6.5 

+   4-9 

1870 

+   3-4 

+  u.(, 

3.2 

+  4.6 

+   7-2 

2.6 

-1-  4." 

+  6.9 

+   5-4 

1871 

+   5-4 

+    S.2 

2.8 

+   5-1 

(-   7-8 

2.7 

+   5.2 

+   8.0 

+  6.6 

1S72 

+   6.0 

+   8.7 

2.7 

+  (1.2 

f    <)■(! 

3-4 

+  6.1 

+  9-2 

+  7-6 

1873 

+   6.9 

+   f)-4 

2.5 

+   6.9 

+  10.2 

3-3 

+  6.9 

-1-I0.2 

4-  8.6 

1874 

+   8.1      +11. 4 

3-3 

+  7.t 

>  lO.S 

3.7 

+  T.(> 

+  M.I 

+  9.4 

'J'he  last  column  exhibits  the  apparent  tabular  errors  in  mean  right  asccnsicii,  and 


13 


therefore  iii  mean  longitude,  ;is  tlorivfd  cacli  vt-ar  fnun  all  Hit-  nhscrvatiuiis.  Tiic  siiddfii 
appaicid  alleralioii  oi'  iicaily  oiio  second  per  aiiiiiiiii  in  (lie  mean  motion  ol'  the  moon, 
exiiihited  in  tiiis  eohimn,  si'ems  t<»  me  oni;  ol'  the  most  (jxtraordinary  of  astronomical 
phenomena;  bnt,  as  I  have  discussed  if  in  sevi'ral  |»a|iers  during  the  last  live  years,  I 
siiall  Ju  no  more  here  tlian  call  attentitm  lo  its  continuance,  ami  to  the  inipossihility  of 
representing  it  by  any  small  mindter  (d  periodic  terms  without  introducing  discordances 
into  tlie  longi(n<le  during  previous  years. 

It  will  he  seen  that  there  arc  discordances  hetwetMi  the  resulls  of  the  two  oi)serva- 
tories,  sinnetimes  aiiKvuiilinL'  to  more  than  a  second.  In  delermining  the  correctitMis  ol" 
short  period,  it  is  desiralde  to  reduce  the  systennitic  erntrs  exteinling  through  each 
year  to  a  minimum ;  the  <[uestion  whelher  such  error.s  arc  in  the  theory  or  the  ol».serva- 
(ioiis  being  indillerent.  It  is  also  desirable  that  in  taking  the  mean  of  the  r<'sults  (d'the 
two  (djservatories,  they  should  be  nuide  comparable  with  each  other  by  correcting  either 
of  them  for  the  .systemaiic  dillercnce.  'I'he  e  corrections,  of  course,  oidy  admit  of 
approximate  determination,  and  they  have  been  applied  ea(di  year  to  (hat  observat(»ry  or 
that  limb  of  tin-  mocm  in  which,  judiring  from  the  deviations  from  unilltrm  proirn-ssion,  it 
was  jiidgt'd  most  likely  that  the  discordance  existed.  The  following  are  (he  correcdons 
actually  applied  to  the  .<evend  clas.><es  of  tabular  errors: 


niLX'invii 

Ii. 

I. 

II. 

s. 

s. 

0.06    1 

+    0 .  of) 

P 

0 

0 

0 

1) 

1-  O.oO 

o.oO 

0 

0 

0 

0 

0 

0 

0 

Waslii 

ngl 

on. 

I. 

II. 

s. 

•  s. 

0 

<  f.Oi^ 

0 

(J 

0 

—   0.04 

0 

—  0.04 

0 

—  0.04 

0 

0 

0 

—  0.04 

0 

0 

Ifavin;.' ap|tlied  th«>se  C(»rrec(ioiis  throughout  their  s«;veral  yt.-ars,  the  Greenwich 
and  Wasliiuirton  ob.servations  were  considered  s(riclly  comparable;  and  when  (he  mtH)ii 
was  (d»served  a(  l»oth  oli.^erviitories  on  the?  same  day,  (he  mean  of  (he  correc(ed  (abular 
errors  was  (aken.     Tin-    meiin   ou(s(ainlinir  (abular  error  \\)v  each  vear  now  becomes  as 


oUows : 

Vear. 

,'? 

Vear. 

.1/ 

YuiU'. 

(P. 

Year. 

-!?. 

1S62 

—  2.1 

1S66 

-1    2.2 

1S69 

+  >' 

1872 

+  7-:, 

1S63 

—  0.9 

1N67 

+  J.^ 

1870 

+  .v6 

1^7.5 

4-  8.0 

1S64 

+  0.4 

1868 

+  4-1 

1871 

4-  6.6 

1S7.1 

+  9-7 

1S65 

+  '4 

'J'hese  quantiti*'.*!,  with  the  sign  <'lianged,  siioidd  b(!  considered  as  c(nrec(ioiis  (o  the 
tundameidal  argument,  and  we  have  (<»  de(ermine  (he  corresponding  correction  (o  (h(.> 
right- a«'"?ii.sions  which  are  (o  be  applied  (o  (he  individual  tabular  ern  s.  To  reduce 
(hem  ti.  ."iM'ri'ctions  o("  true  Innirituile,  (hey  are  (o  be  multiplied  by  the  factor 

I  +  -  ''  •■^•'^  i'  =^  •  +  o.  1 1  cos  i; 


14 


Tin;  (roiTcspoiuliiiff  taclor  tor  correct  ion  of  riglit  ascension  is,  witli  siitticient  ui)]>rox- 
iniiition, 

Sazn  {i  -\-o.i  \  CDS  i,'  —  0.04  cos  (2  /  —  0)  —  0.09  cos  2  /)  SX 

In  this  I'orninlu,  <5/V  represents  tlie  correction  ti>  the  mean  loiif(itn»ie,  while  we  may 
sii|)i)ose  /  to  represent  indillerentl}'  the  nu-an  or  tlie  tru(!  lonyitiule ;  and,  during  a  period 
ol'  several  months  at  a  time,  we  in.-iy  represent  tht;  lonijilnde  as  a  t'unction  of  g.  The 
valu(!  of  Sa  has  Ih'.cm  reduced  to  a  table  of  doultle  entry  as  a  function  of  if  and  of  tlie 
time.     To  express  th(!  mean  longitiule  as  a  function  of  if,  we  have 

/-        i/+        TT 

2  /  —  0  zz  2  g  -[-  2  ?r  —  0 
I  7—  2  g  -\-  2  7r 

My  the  substitution  of  these  values,  the  expression  tiir  Sa  becomes 

(5a  ^;  ( I  -f-  o.  I  I  COS  g  -\-  A  COS  2  g  -\-  li  sin  2  g)  S\ 
w  lie  re 

A  ZZ  —  .04  COS  (2  /T  —  0)  —  .09  cos  2  /T 

li  ■=.      .04  sill  (2  /T  —  0)  -f-  .09  sin  2  TT 
The      lues  ot   rr,  0,  A,  and  li  \\)V  periods  of  six  months  are  as  follow  : 


of  tl 

spoi 

this 


Yoar. 


1862.0 
j  1862.5 
j  1863.0 
1863,5 
186.). o 
1864.5 
1S65.0 
1S65.5 
I S66 . o 
1S66.5 
1867.0 
1S67.5 
18OS.0 
186S.5 


The  coeftl( 

lest!   sets  of 
idiiijr  valin;  < 
piipcr,  it  is 
The  correct 


T 

() 

„ 

22S 

274 

2-lS 

264 

269 

255  1 

2S() 

245  , 

3'-iy 

235 

33'-> 

226 

.350 

216  ] 

310 

206  1 

31 

"J7 

51 

1S7 

71 

■77  1 

.)2 

168  j 

112 

15S  1 

»33 

14S 

+ 
+ 


.05 

.0(| 

.oS 

.03 
.02 

•  05 
.06 

■  05 
.ot 
.02 

.05 
.05 
.04 
.03 


B 


+ 
+ 


.0(J 

.01) 
.04 

.oS  I 

■  07  ! 
.04  ^ 

.00  i 

.03  ; 
.03 

■  "5 
."3  ' 
.00  t 

.02  I 

.05  i 

I 


Year. 


18O1J.0 
1S69.5 
1870.0 
1870.5 
1S71.0 
1S71.5 
1872.0 
1872.5 
1S73.0 

1S73.5 
1874.0 

1S74-5 
1S75.0 


JT 

« 

„ 

• 

153 

•39 

"73 

129 

";4 

119 

214 

no 

234 

KX) 

255 

90 

275 

81 

295 

71 

316 

61 

33f> 

52 

SSf" 

42 

17 

32 

37 

23 

.07 
.08 
,06 

.01 

.06 

.  10 

.09 
.04 
•05 

.12 
,12 
.04 


B 


.06 
.05 

,1K) 
.05 
.09 
.08 
.02 
.06 
.11 
.11 
.04 
.05 
,  12 


ient    1  +  O- 1 '  t'"*'*  rJ"  +  ^  cos  2  g  -{■  li  sin  2  g  is  next  tabulated  for  each 
values  of  A  and   li  for  every  10  '  t»f  g,  and  multiplied  by  the  corre- 

if  iiX.      As  these  tables  are  superseded  by  those   yivcii  at  the  cIos(!  of 

not  necessary  to  print  them, 
ons  ol"  short   period,  which  have  been  actually  a|»plied,  are 

-I-0.96  sill  1) 

—  0.13  sill  {D  -\-  g') 
+  0.09  sill  g' 

—  0.62  sin  (2  i'  —  4  5^'  +  2  6j  —  4  (x)') 


15 


The  first  tliroo  liave  been  combinod  info  a  siiiifl(>  oih;  olMoiihlo  aii,niiii(Mit,  in  wliidi 
tlie  argunionts  sire  />  antl  tlio  niontli;  Uk!  liitliM-coiic'spontlinij;  to  i('.  Tlic  Icinis  dcpcnd- 
ent  on  tiiis  argument  nw.  so  small  that  they  may  l>e  regarded  as  eonstant  during  an 
entire  month. 

In  |,his  sain(>  talde  is  ineludcd  a  partially  conjectural  correction  l()rtlit!  variations  (tf 
the  moon's  semi-diameter.  The  correction  to  Hansen's  value  has  l)eeii  assumed  a.-; 
—  2".o,  when  the  moon  is  in  the  ntnghhorhood  of  the  sun,  so  that  iier  limb  is  very  liiint; 
and  as —o".4  after  the  dose  of  evening  twilight.  IJetween  two  hours  of  elongation 
and  the  dose  of  twilight,  it  is  assumed  to  increase  uniformly.  The  sum  ol'  these  l()ur 
corrections  is  given  in  the  tbllowing  table : 


0)  — 
Ul    t,    O 

cT  S  S 
Q 


FIRST  LIMR. 


Jan. 


14 

+  2.4 

13 

+  2.3 

12 

+  2.2 

II 

+  2.1 

10 

^■  2.0 

9 

+ 1.8 

8 

+  1.5 

I 

+ 1.5 

fi 

+ 1.5 

5 

+ 1.4 

4 

(- 1.2 

3 

+  1.1 

2 

+  0.() 

1 

+  o.f) 

0 

+  0.4 

Full 


+  2.5 
+  2.4 
+  2.3 
+  2.2 
+  2.1 
+  2.0 
+  1.7 
+  1.5 
+  1.4 
+  1-3 
+  1.2 
+  1.0 
+  o.S 
+  n.6 
+  0.4 


Mar. 


+  2.5 
+  2.5 
+  2.5 
+  2.4 
I-  2.4 
+  2.3 
+  2.1 
+  I.S 
+  1.4 
+  1.3 
+•  1.2 
+  I.O 

+  O.S 
+  n.6 
+  o.. 


.•\l)iil.  !    May.   I    liiiu'. 


+  2.6  !  +  2.5 
-f-  2.5  I  +  2.4 


+  2.4 
I  2.4 
+  2.3 
+  2.2 
+  2.0 
+  1.8 
+  1-5 


:.4 
2.3 


+ 
f 

I-  2.2 
+  2.1 
+  2.0 
+  I-S 
+    I.f. 


-H  1.2    i  +  1.4 

+  I  .  I  I  t-  I  .  I 


+  1.0 

■1-0.  (J 

+  0.8 

+  0.7 

+  0.6 

+  0.6 

+  0.4 

+  0.4 

+  2.4 

+  2.3 
+  2.3 

-I-  2.2 
\-  2.1 
+  2.1 
+  2.0 
+  1.8 
I--  1 .  5 

+  1-4 
H-  I.I 
+  ".'J 
+  ".7 
+  0.6 
+  0.4 


Inh. 


Aug. 


Sept. 


-I-  2 . 3  '  -I-  2 . 2  +2.1 

+  2.2     4-2.1  +2.0 

(-2,24-2.1^  +  2.0 

■f     2.1     I     f-    2.1  +   2. CI 

1-  2.0    '   -f-   2.0  I-  2.0 

+  2.0     4-1.1)  4  I.S 

-I-  i.()     4-  I.S  4-1.7 

4-  1.7  I   t-  1.6  I  4-  1.4 


4  I. ; 
4-  I.I 

4-  I . ' 

4  11.9  ;  4-  1.0 


+  1-5 
+  1-3 
t-  1 .0 


I 


-ho.S 

4- o.S 

-f  0.6 

4-  0.6 

-f  C.4 

4-  0.4 

4-  1.2 
+-  1.2 
4-  I.I 

f  1.0 

-f  0.8 
-I-  0.6 
4-  0.4 


Oct. 

Nov. 

Der. 

+  2.1 

4-  2.2 

-!■  2.3 

4-  2.0 

4-  2.1 

•t-  2.2 

+  1.0 

■(-  2.0 

4  2.0 

1   1.8 

-1-  I..) 

4  2.0  1 

i  1.7 

1-  1 . 7 

4-  I.S 

4-  1.6 

4-1.5 

4-1.6 

4-  1.4 

+  1.4 

+  1.5 

+  1-3 

4-1.4 

+  1.5 

+  1.3 

4-  1.4 

4-  1.4 

+  1.2 

4-  1.3 

-1-  1.4 

4-1.1 

4-1.2 

4-  1.2   1 

4  1.0 

f  l.I 

4  1.1 

4-0. S 

+  o.<) 

4-  0.9 

4-  0.6 

4-0.7 

4  0.6 

40.4 

-ho.4 

4-0.4 

"^   r!   O 
lyi   4j   O 

S'S  s 


c  c 

IT     O    O 

Q 


4 
4- 
-I- 
+ 
4- 
+ 
4- 
+ 

4- 
+ 
4- 
+ 


o 
I 
2 
3 
4 
5 
6 

7 

S 

9 
10 
II 
1 2 
13 
14 


Jan.    I    Feb. 


-0.4 

-0.4 

-0.6 

-0.6 

-O.S 

-0.7 

-I.I 

-0.9 

—  1.2 

—  I.I 

-  l;4 

-1.2 

-  1.4 

—  1-3 

-  1.5 

-  1.3 

-  1.4 

-  1.5 

1.7     ;    -    1.7 

I 


—  1.9 
-    2.0 

—  2.1 

—  2,2 
-2.3 


—  1.9 

—  1.9 

—  2.0 

—  2.1 

—  2.2 


Mar. 

April. 

-0.4 

-0.4 

-0.6 

—  (1.6 

-o.S 

-o.S 

—  l.O 

—  1 .0 

—  1.2 

—  I.I 

—  1.2 

—  1.2 

-  1.3 

-  1.4 

-  1.6 

-  1.6 

-  I.S 

-  I.S 

-2.0 

-1.9 

—  2.0 

—  2.0 

—  2. 1 

—  2.0 

—  2.1 

—  2.0 

—  2.1 

—  2.0 

—  2.t 

—  2.1 

Mav. 


-  ".4 

-  0.6 

-  (I.S 

-  0.9 

-  I.I 

-  1.4 

-  1.5 

-  1.7 

-  1.9 

-  1-9 

-  2.0 

-  2.0 

-  2.1 

-  2.  I 

-  2.2 


s 

FCOND  LIMB 

Dec. 

June. 

July. 

Aug, 

Sept. 

Oct. 

Nov. 

-0.4 

-0.  t 

-0.4 

-0.4 

-0.4 

-0.4 

-0.4 

-0.6 

-  0.6 

-0.6 

-0.6 

-0.6 

-0.6 

-0.6 

-o.S 

-0.7 

-0.7 

-o.S 

—  o.S 

-o.S 

-0.8 

-0.9 

-0.9 

-0.9 

-  1.0 

—  1 .0 

—  1.0 

—  I.I 

—  1.0 

—  1.0 

—  T.I 

—  I.I 

—  1.2 

—  1.2 

—  1.2 

-  1.4 

-  1.4 

—  1.2 

-  1.3 

-  1.3 

-  1-4 

-  1.4 

-  1-5 

-  1.5 

-1.5 

-  1.4 

-  1-4 

-  1.4 

-  1.5 

-  1-7 

-  I.S 

-  1-7 

-  1.6 

-  1.5 

-  1.5 

-  1-5 

-  1-9 

—  2.0 

-  1.9 

-  1.9 

-  1.7 

-  1.5 

-1.5, 

—  2.0 

-  2.1 

—  2.1 

—  2. 1 

-  1.8 

-  I.S 

-  1.7 

—  2. 1 

-2.1 

—       '*       2 

-  2.3 

—  2.1 

—  2.0 

-  1-9 

—  2.1 

—  2.2 

-2.3 

-2.4 

22 

-  2.2 

-  2.0 

—  2.2 

-2.3 

-2.4 

-  2.4 

-  2.4 

-2.3 

—  2.1 

-2.2 

-  2.3 

-2.4 

-  2.5 

-2.5 

-  2.4 

-2.3 

-  2.3 

-2.4 

-2.5 

-2.6 

-2-5 

-2.5 

-2.4 

-  14 

-  '3 

-  12 

-  II 

-  10 

-  9 

-  8 

-  7 

-  6 

-  S 

-  4 

-  3 

-  2 

-  I 
o 


c  c 
«  o 

(/I  01   o 

J?E  6 

a 


o 
I 

2 

3 
4 

5 
6 

7 

8 

9 
10 

II 
12 
13 
14 


-f- 

4- 
+ 
I- 
+ 
4- 
4- 
,  -t- 
-I- 
4- 
■I- 
4- 
+ 
4- 


■  •*'=Wrtfe«*rtSS«'HilBBWl»Slj*.i«  0tmy**'-j. 


>^i.i**wm^3«w**r' 


i.  I 


I  ll 


ij'  111 


«tl 


16 

]\y  tlic  ii|)|)li(;i)tit)ii  of  tlio  foroiroinir  corrections  to  tlic  errors  of  llic  moon's  taluilar 
rii^lit  asf-ensioii,  these  errors  iiiiiy  he  supposed  to  he  rediictMl  to  very  small  (piaiitities, 
dependiiiij;  on  the   errors  of  the  lunar  (!l(!ments,  with  which  th(;y  are  eonnecleil  hy  the 

e(piation 

„       ■    (lot   ^,    ,      (In  »,j    ,     (la  ^. 
(U        ^    (1.0        ^    (li      ' 
the  clillercntial  coelticieiits   having  tlie  vahuvs   ijiven  on  |)aire  12.      When  we  snl)stitnte 
theso  values,  tiie  exprcission  for  Sa  will  contain  the  terms 

(+  .01 8  (5(9  —  ,03 7  '5a)  cos  (2  /  —  <9) 

—  .087  6a  cos  2  / 
+  .018  (5  (9  cos  (9 
-|-  0.21  <5/  sin  6 

—  0.2 1  (3i  sill  (2  /  —  0) 

If  we  represent  the  sum  of  these;  terms  l>y  P,  we  shall  have 

SI  =<'ia  —  'D 

In  the  investigation  of  the  corrections  to  the;  moon's  eccentricity  and  longitude  of 
perigee,  the  terms  of  P  may  be  entirely  neglected.  This  arises  from  the  circumstances 
that  tlu!  appreciable  terms  of  /or  a  arising  tV'im  tlie  errors  of  these  elements  liavt;  the 
same  period  with  f,  tiw;  mean  anomaly,  while  /'contains  no  apprecialde  pi'riodic  terni 
depifuding  on  g.  The  outstanding  pmtion  of  ('ia  prol)aldy  averages  not  more  Ihan  »me 
second  or  two  at  the  utmost,  so  that  the  term  .037  ('>a  is  (piite  insignificant.  The  term 
.018  SO  may  have  a  constant  value  of  o".25,  more  or  le.ss;*  l)ut  tlu;  short  period  of  the 
term  2/ — (9,  ami  its  inc(»mmensnral»ility  witli  the  period  of  i,',  permit  of  this  error 
i)eing  regarded  as  liirtiiitoiis.  The  sanii'  rentark  ap[»lies  to  tiiiMcirms  .0S7  S(t  cos  2/ 
and  0.21  r5/sin(2/ — 0).  'i'he  only  remaining  terms  liave  tiie  jteriod  of  0,  which  is 
niorf-  than  (Mghleen  ye:irs.  'J'lie  ellect  ot'  these  possildt;  errors  is  tlieiell)r<'  eliminated 
in  th(>  mean  correction  for  each  year,  which  has  been  alrea<ly  applii-d  to  the  errors. 

To  determine  the  correction  to  tin!  (•(•(•entricity  and  longitude  of  the  perigee  result'- 
ing  from  each  year's  observations,  the  residuals  in  riglit  ascension,  afl(!r  the  application 
of  the  three  corrections  already  described,  have  becjn  arranged  according  to  the  values 
of  the  mean  anomaly  to  which  they  corresi»ond.  The  results  are  shown  in  the  follow- 
ing talde,  which  gives  for  certain  limits  of  mean  anomaly  in  the  first  ctdumn,  firstly,  the 
sum  of  the  residuals  (tal>idar  iiiliiii^  o!)served)  in  riglil  asctiusion,  corresponding  to  all 
the  values  of  mean  anomaly  between  those  limits;  and,  secondly,  the  number  of  the 
residuals.  In  taking  these  sums,  the  observations  at  the  two  ob-servatories  are  counted 
st^paratrly,  .so  tl'.at  when  observations  wen;  made  at  \m\\\  obs(M'vafories  on  the  same 
date,  th(!  sum  of  the  residuals  is  tak(Mi,  ami  the  observations  count  2  in  the  column  N. 

"It  IH  iiftcrwiinl  tuiinil  that  the  viilno  of  this  |iii>.     ■■t  is  only  o  .oS. 


17 

Sums  of  errors  of  tnoo)i\s  corrected  rifjht  ascension,  given  hy  ohserrations  at  Grccnmch  und 

Washington. 


1362. 

1863. 

1864. 

1865. 

Limits  of  mean; 







anomaly. 

i 

1 

N. 
f 
1 

£.!a 

N. 

S,!n 

N. 

D.ia 

N. 

. i 

0                        o          i 

o  lo     lo 

+     3-9 

4 

H 
+      21.5 

10 

+    I9.f> 

9 

+     1-4 

7 

10   lo      20 

+     3.f' 

6 

+  12.3  : 

12    ! 

+     6.1 

7     : 

+     3-4 

4     * 

20   to      30 

—     0.2 

5 

+  14.2 

8 

+     5-8 

5     j 

—     <i? 

10 

30  to     40 

+     9-3 

8 

+  23.7 

II 

+     4-5 

7 

-     0.5 

5     1 

40  to     50     1 

+     2-7 

8 

+     9'0 

8 

+     2.6 

3 

-      3.f' 

6 

50  to    60 

+     0.3 

8 

+     9-8 

9 

-     1.6 

10    1 

—     I.I 

6    1 

60  to     70     '■■ 

+     S.9 

10 

-     4-3 

7 

+     '-.7 

5 

-     61 

7 

70  to     So 

-     3-7 

4 

+     7-0 

10 

-     7.0 

f'     i 

-    f>.7 

6 

80  to     90 

+     6.7 

7 

-     6.7 

6 

—   II. 2 

9 

-    6.1 

6 

qo  to  too 

+     3-9 

6 

-     3-3 

9 

-     3-4 

6 

-     8-5 

7 

too  to  ito 

+     3-9 

11 

-     0.4 

5 

-     2.1 

5 

—     0.7 

5 

110  to  120 

-     6.4 

9 

-     3.9 

8 

-     30 

3 

-     7-5 

8 

120  to  130 

-     3-2 

8 

-     3.9 

7 

+     0.1 

5 

-     5-5 

6 

130  to  140 

-     7-8 

6 

-     8.8 

8 

—   12.2 

7     i 

+     5-0 

5 

140  to  150 

-     0.9 

5 

-   15-9 

8 

+     0.9 

3     1 

i 

+     ... 

5 

150  to   lf)0 

—     0.1 

5 

-    18.2 

9 

-     f>.7 

7 

+     1.5 

4 

160  to  170 

-     8.8 

4 

-  19.7 

6 

+     2.5 

6 

+     4-3 

5 

170  to  iSo 

-     5.7 

4 

-     9-9 

7 

-     5-3 

5 

+     6.4 

6 

iSo  to  190 

-  17-4 

9 

-  33.1 

14 

-     8.6 

7 

+     8.9 

6 

190  to  200 

-  15-5 

7 

-     4-3 

4 

-     0.6 

4 

+   15-2 

8 

200  to  210 

-     3.S 

10 

—     1 .0 

6 

-     6.4 

9 

+     7-8 

8 

210  to  220 

—     0.2 

2 

-     1.9 

9 

-     2.9 

8 

+  13  I 

7 

220  to  230 

—   28.9 

9 

-     7.5 

10 

+     3.<i 

/ 

+     5.1 

5 

230  to  240 

-     7-3 

7 

-     1-9 

7 

+     0.8 

7 

+  10.3 

5 

240  to  250 

+   I3-0 

8 

+     0.4 

9 

+      1.6 

7 

4      7.3 

8 

250  to  260 

—     2.0 

4 

+     7-6 

8 

+   II.5 

8 

+     7-3 

7 

260  to  270 

+     1.6 

9 

+     1.4 

!      5 

+   11-7 

7 

+   16.2 

12 

270  to  23o 

+     3-7 

5 

+  II. 3 

9 

'     +  25-3 

II 

+     7.(' 

II 

2S0    to   2</3 

+     4-7 

7 

1-     o.S 

5 

+   18.2 

8 

+     9-f' 

8  : 

290  to  300 

-     1.3 

1       I 

+   15-9 

7 

;     +     6.6 

4 

+     5.8 

II  1 

300  to  310 

+     3-0 

i      3 

+  23.5 

9 

+     7.8 

6 

+   10. 1 

7  i 

310  to  320 

+     2.3 

1       2 

+    22.6 

;    6 

+     6.4 

5 

+   16.4 

10 

320  to  330 

-     2.8 

5 

+    18.2 

9 

+   11.6 

7 

\     +   14.5 

7 

330  to  340 

+     9-5 

6 

+      1.2 

7 

+   18.5 

10 

+   16.7 

II 

340  to  3;o 

+   II. 8 

8 

+     7-2 

7 

+     4-2 

7 

+     7-6 

1 

7 

350  to  360 

+   i3.f' 

5 

+   14-4 

8 

+    16.5 

6 

!      +      5-3 

9 

j     +106.4 

225 

+  222.0 

2S7 

1     +1S7.1 

i  236 

;   +205.9 

255 

i     —116.0 

1 

1 

1 

-144-7 

- 

1     -   71.0 

j 

-  46.6 

- 

j     -     9-6 

+  78.3 

+  116. 1 

+  1593 

Sm 


»^»*>*~-"'*«»e!B«»«wss*<i«nBi4ii«»*'»» 


18 


Slims  of  erron  of  7noons  corrected  right  aKcension,  S^v. — Contimiod. 


1866. 

1867. 

1868. 

i86g. 

Limitsof  mean 

anomaly. 

£(ln 

N, 

Ida 

N. 

XAa 

N. 

2, In 

N. 

0  to     10 

-     1.7 

6    i 

+      7.4 

5    ! 

n 

-   4.2 

4 

-    10.7 

4 

lo  to     2o 

-     2.5 

4 

-      5.0 

2    1 
i 

+     3-9 

7 

-     4.2 

4 

20  to     3P 

-     7.5 

3 

-      1.7 

4 

-     2.5 

3 

-     0.8 

6 

30  to     40 

-     7.1 

5 

-      7.5 

3     , 

-     9-4 

6 

+     4.2 

5 

40  to     50 

-   14-5 

7 

+      5.5 

' 

-     g.o 

5 

+   II. 0 

6 

50  to     60 

-     0.7 

I 

—     2.0 

4 

-     0.7 

7 

+     5-5 

3 

60  to     70 

+     1.3 

5 

-     8.5 

4 

+      2.2 

7 

+     3.1 

5 

70  to     80 

+     5-3 

6 

-     4.8 

3 

+     4.1 

8 

+     7.7 

7 

80  to     go 

+     1.6 

6 

-     3.6 

I 

+    12.2 

7 

+     8.0 

8 

90  to  100 

+     3.9 

4 

+       2.6 

5 

-      0.3 

4 

+   16.8 

8 

100  to  no 

+     4-4 

9 

-     0.6 

5 

+    14.9 

7 

+     5-1 

9 

no  to  120 

+     4.« 

8 

+     3.9 

5 

+      9.8 

6 

+     8.3 

6 

120  to  130 

-     5-4 

8 

-(-     1.6 

7 

+      4.1 

5 

+  14.5 

7 

130  to  140 

+     3-4 

6 

+     4.1 

6 

+    10.2 

8 

+     7.5 

8 

140  to  150 

+   lo.l 

9 

+     1.9 

7 

+       5.2 

7 

+     3.1 

6 

150  to  I  Go 

—     4.' 

6 

-     2.6 

7 

+      2.1 

9 

+  20.3 

7 

t6o  to  170 

+     3-3 

7 

+     6.8 

5 

+       ..3 

8 

+     3.7 

3 

170  to  180 

—     0.1 

7 

-     5-0 

8 

+    0.8 

7 

+  12.2 

7 

180  to  190 

+     o.S 

6 

-     0.3 

2 

+  12.3 

8 

+     7.0 

5 

190  to  200 

+     5-9 

6 

+     2.0 

4 

+  17.9 

6 

+     6.3 

4 

200  to  210 

-     3-2 

6 

+     2.8 

6 

+     5.2 

5 

+  10. 1 

5 

210  to  220 

+     0.3 

6 

-     1-7 

4 

:  +  13.0 

8 

+    12.2 

5 

220  to   230 

-     5.4 

4 

+   12. 9 

9 

1  +  4.8 

4 

+    12.3 

7 

230  to  240 

+     4.J 

8 

+     8.2 

6 

+    15-2 

9 

-       1.3 

3 

240  to  250 

-     1.8 

7 

+  25.4 

9 

i  4-     7.4 

8 

-      6.4 

6 

250  to  260 

+     9-4 

'       7 

+     0.9 

3 

'.  +   14.2 

8 

-      3.6 

2 

260  to  270 

+     2.7 

7 

+    It. 7 

6 

:  -    5.0 

2 

-  >7.3 

7 

270  to  280 

+    9-: 

4 

;  +   3-3 

4 

+       I.O 

7  • 

-  18.8 

5 

280  lo  290 

+   II. 6 

12 

1 

+     7.0 

7 

-   9.1 

5 

-  21.4 

6 

290  to  300 

+     4.0 

!  4 

+    0.7 

3 

-     3.2 

8 

—  13.6 

3 

300  to  310 

+     6  7 

4 

+  16.5 

7 

-    8.0 

i      2 

-     4.8 

2 

310  to  320 

+     3-4 

2 

i  +     2.3 

5 

-  13.8 

8 

-    0.8 

I 

320  to  330 

+     7-7 

5 

1  +    0.2' 

5 

—   10.6 

9 

-     4-2 

2 

330  to   340 

+     9-1 

5 

1  +     3.5 

6 

-  11.7 

6 

-  18.5 

6 

340  to  350 

+  10.8 

6 

\  -     5.4 

7 

.  -     9-8 

I 

5 

—   10.6 

4 

350  to  360 

+     92 

7 

-     7.2 

4 

1  -   18.3 

6 

—     2.2 

5 

+  132.9 

213 

+  131. 2 

182 

+  161. 8 

229 

+  178.9 

187 

-   54.0 

1  -   55-9 

—  115.6 

1 
1 

-139.2 

+  78.9 

1 

1  +  75.3 

+  46.2 

+  39-7 

19 


Sum/)  of  crwrs  of  moon\s  corrected  right  ascension,  S^r. — ('(mcliidcil. 


Limits  of  mean 
anomaly. 

1871 

187 

. 

1872. 

1873. 

1874. 

7,,\a 

N. 

Sila 

1 

2,1(1 

1 

N, 

V 

rta 

N, 

S,l<. 

1 
N. 

OlO      10 

-     7.2 

5 

-     3.2 

5         +     6.5 

6 

4.3 

6 

+     4.6 

1 
1 

1      '* 

10  to     20 

—      2.2 

5 

+     1.7 

n    1    +    8.5 

10 

+ 

5.2 

4 

+     5.9 

S 

20  to    30 

+     5.1 

6 

-     0.3 

7 

+     5.5 

8 

+ 

5.2 

8 

+   12.5 

i      f' 

30  to   40 

-f   10.7 

8 

+     6.4 

7 

+  1 1. 8 

7 

+ 

3.4 

3 

+     5.1 

i       5 

40  to    50 

+    II. 3 

8 

+   16.7 

9 

+     6.0 

4 

+ 

6.6 

4 

+     4.4 

i 
j       5 

50  to    60 

-       7.1 

5 

+     9.7 

6         +   13,2 

6 

+ 

4.1 

7 

+      2.1 

5 

60  to    70 

+       I.O 

9 

+   18.9 

8         +   10.4 

3 

+ 

13.4 

6 

+    10. 1 

4 

70  to    80 

-      2.6 

5 

4-    10.2 

7    1     +  12.4 

8 

+ 

'3.5 

3 

+     6.6 

6 

80  to    90 

+  12.0 

12 

+  n.7 

1      5     *     +  II. 3 

4 

+ 

15.8 

7 

+     6.0 

3 

90  to  100 

+    10. 1 

8 

+  12.5 

i      3 

+     9-8 

4 

+ 

5.1 

2 

+     5.9 

7 

100  to  no 

+  10.8 

4 

+  19.7 

8 

+   13.0 

6 

+ 

1.5 

3 

+   10.9 

6 

no  to  120 

+     5-8 

6 

+    8.2 

4 

+   18.7 

6 

+ 

5.3 

2 

+     4.6 

4 

120  to  130 

+   10. 1 

7 

+    9.7 

5 

+   18.3 

7 

+ 

6.1 

5 

-     4.7 

6 

130  to  140 

H-   10. 1 

5 

+  «5.4 

5 

+     0.2 

2 

+ 

3.3 

3 

+     1.8 

1 

140  to  150 

+   18.2 

8 

+     2.1 

3 

+     2.9 

3 

+ 

8.4 

5 

-     0.8 

7 

150  to  160 
160  to  170 
170  to  180 
180  to  190 

+    4.4 

+■     8.8 
+.   6.9 

+     I  8 

3 
5 
3 
I 

+     3.0 
+     8.7 

+      6.2 

+     3.9 

7 
4 
6 

+     2.1 
+     6.6 
—      1.2 
+      1.9 

8 
5 
3 

4 

- 

3.9 
5-4 
1.7 
2.2 

3 
4 
3 
4 

+     1.3 

—  10. 1 

—  1.0 

+     5.0 

5 

9 
6 

6 

190  to  200 

+     7.5 

4 

+     3.5 

3 

—      1.2 

5 

- 

6.6 

6 

—           1.0 

2 

200  to  210 
210  to  220 
220  to  230 

+     2.1 
-     2.5 

5 
3 

3 

+     1.0 

-  2.6 

-  9.3 

3         -     2.2 
2         +1.2 
7-7.2 

6 
3 

5     1 

0.9 
6.6 
o.i 

2 

3 
4 

+     3.7 

-  5.0 

—  16.0 

6 

5 
7 

230  to  240 
240  to  250 

250  to  2f)0 

260  to  270 
270  to  280 
280  to  290 

-  0.4 

-  9-7 

-  12. I 

-  2.3 

1 
-.2.9 

-  5.6; 

5 
5 
6 
2 
8 
3 

-  3. a 

-  9.1 

-  5.2 

-  4.6 

-  7.1 
--    2.7 

6  -     4.8 

8       ;       -       6.5    ^ 

5          -     9-1   i 

5  i     -  13.8 

7  ;     -     8.4  \ 

6  t     —   16.7 

5     ' 
3    : 
4 
8 

5 
9 

— 

3.5 
7.5 
7.1  1 
8.6 

4.3  1 
10.8  i 

I  ! 
5  ' 
4 

1 

3  i 

4  1 

6 

-  «3.5 

-  15.1 

-  23.0 

-  22.6 

-  I?. 6 

-  9.1 

4 

8 
5 
4 
4 
3 

290  to  300 

-       5.5 

4 

+    4.0 

4     !     -   10.3 

S 

— 

9.8 

4 

-    13.4 

S 

300  to  310 

-      4.0 

4 

-    9.5 

6-9.5 

5 

— 

1.8  1 

I 

—     O.I 

9 

310  to  320 

-      8.7 

3 

-    6.6 

5     i     -     5-6 

4 

— 

3.2  ( 

4     1 

-    ;.3 

6 

320  to  330 

-    13.5 

6 

-     4-9 

7     i     -     8.5 

5 

— 

11-3 

7 

-    1.4 

7 

330  to  340 

-     ..7| 

4 

-     2.8 

7     1     -     8.5  : 

5 

— 

0.3 

3 

-    4.3 

3 

340  to  350 
350  to  360 

-  3.6  i 

-  8.7  ! 

3 

5 

-     1.7 

+     6.3 

4 
4 

-     5.1  ' 
+    0.1 

5 
6 

— 

9.2  '■ 
4.0 

6     I 
5     1 

+    2.2 

+     2.5 

II 

6 

+  136.6  , 

185 

+  179.5 

203 

+  160.4  i 

195 

+ 

56.9 

155     . 

+  95.2 

200 

—  120.6 

-  72.8 

-118. 6  j 

-113. 1 

— 171.0 

+   16.0 

+  106.7 

i 

i     +  41.8 

- 

16.2  '■ 

1 

-  75. s 

Neglecting  all  tcrinr,  multiplied  hy  the  eccentricity  in   tlu;  coefficients,  each 
ual  gives  an  equation  of  the  form 

Jl-\-  2  sin  gJe—  2  cos'g  e  Jtt  —  r 


rcsKi 


'"•^^BBimMn 


■  r 


1' 

1!i 


■'! 


II- 


or,  piittirii, 


the  ncjuatioii  will  \hi 


20 


h  zz.  2  JSe  :r  —  2  ih' 
k  z=.  —  2  J(i  y')Tr  ■=:  2c  Stt 


Jl  -\-  h  ^\x\  g  -\-  k  ros  if  ■=.  )\ 


Je  nnd  Jit  I)i;ing  tlio  errom  of  tlio  taimlar  eccentricity  and  longitude  of  the  perigee, 
while  8e  and  fin  n^proseiit  IIk;  corresponding  corwct'ums. 

The  erpiations  are  now  solved  as  if  all  the  residuals  within  each  pair  of  20°  limits 
corresponded  to  the  mean  ol  the  limit, — that  is,  as  if  all  hetween  0°  and  20°  corre- 
sponded to  i'  zi  10°  ;  those  l)etween  g  zz.  20°  and  if  =.  40°  to  g  —  ,30° 


and  so  on. 


If. 


then,  w(!  put 

gi  =  10' 
Vi,  the  SI 


•r-i 


zz  T,o^,  etc. 


)f  all  til 


dual; 


IS  III  any  one  year  corresponding  to  g  =zgi; 
Hi,  the  corresponding  iuinil)cr  of  observations; 
A-.zz  sin  gi] 
Ci  zz  cos  gi : 
the  normal  equations  for  det(!rmiiiiiig  SI,  h,  and  k,  by  least  sciuarcs,  will  be : 
i^'n,)     Jl+(^„,s,)     /t  +  (^'«;6-..)     kzz^u 
(^'  ,U  .V.)  Jl  +  {:^  lU  sr)    h  +  {2  n,  .s,  6v)  k  =  2  Si  t-i 
{2  „i  d)  Jl  +  {^  «,.  .sv  d)  h  +  (^' »,  en    k  =  2  d  Vi 
The   formation  and  solution  of  tli(;se  erpiations   for  each  year  give;  IIk;  following 
values  of  the  oiilstiUKliiig  errors  of  llie  lunar  eleiiienls  l()r  each  year: 


1862,  /<zz  +  o.o4 

1S63,  —0.64 

1864,  —  1.07 

1865,  —  1.03 

1866,  —0.47 

1867,  —0.93 

1868,  +0.34 

1869,  4-  1.67 

1870,  +148 

1871,  +1.65 

1872,  +2.15 

1873,  +1.91 

1874,  +  1.92 
The  periodic  character  of  these  residuals  is  very  reiiiarivable,  indicating,  as  it  does, 

either  a  hitherto  unknown  inerpiality  of  tlie  moon's  mean  longitude,  having  nearly  the 
same  period  with  the  orbital  revolution;  or  one  of  the  eccentricity  and  longitude  of 
perigee,  having  a  period  of  between  tifteen  and  twenty  years.  To  investigate  this  in- 
e(piality,  we  shall  assume  that  each  value  of  h  is  of  the  tbrm 


kzz-Y  1.23 
+  1.78 

+  I -09 
-0.15 
+  0.10 

—  0.36 

—  1.46 
-1.56 
•-  1. 14 

—  0.36 
•     —  o.  1 2 

+  0.16 
+  0.60 


and  each  value  of  k  of  the  form 


h  —  a  sill  (yu  -{-  ut) 
/?;  + a' cos  (yu' +»'/;), 


21 


3S, 

he 
of 
in- 


h  k  a  it'  fi,  fi\  »,  aiul  n'  Iumms,'  m.km.wii  (unuilili.-s  (..  I..-  .Ict.-iiMin.'.l,  an.l  /  tin-  tun., 
in  v,>ars  fmni  unv  assumrd  ..,m.H..  Wr  slmll  tak..  for  tl..-  qM'«'''  tl'"'  "n.l.H.' -'I  t  "■ 
pori.ullhn.UKli  wliicl.tl..-..l.s.Tvati..ns.'xto...l;tliat  is.  1868.5.      If,  tli.-ii,  w  rrpivscnt,  tlw 

thirtiMMi  values  of  //  and  /.•  in  cliroiioloi^ical  order  by  //-,„  /'-.^. ^'u'  /'-r..^'-...  ■  ■  •  •  . 

k„  the  e.umtioMs  .>f  (■on.litioi.  for  h  and  k  resix-etively  may  Ix^  put  into  tlw  lorn. 

h;  zzh  —  a  sill  //  cos  in  —  n  cos  /<  sin  /  n 
A-,.  =  /•+«'  cos  n  cos  /  n  —  n  sin  /(  sin  i  11. 
Rcuanlinii  A,  /.-.  «  sii. //,  a  cos  /^  «'  sin  n,  an.l  «'  cos  /<  as  tin-  unknown  .,nantifi.-s, 
tlic  normal  ecinations  for  determining  thes.;  (piantiti.'s  ar.' : 

(i)   From  the  vului's  of  li,. 
13//  — (2' cos  in)  a  sin /<   —      ^^i 
-  (V OS  /  «)  h  +  (^'  cos-  /•  n)  a  sin  /«  =:  -  :i'  A,  cos  i  n 
(>;"  sin-'  i  11)  a  os  /<  —  —  ^  I'i  sin  '  " 
(2)  From  the  viiiucs  of  A',. 
13/,-  +  {2 cos  i  w)  «'  cos  /<'  =:      ^ ^j 

(^'  cos  i  «)  ^-  +  (^  '''"*'  * ")  «'  ^■"'^  ^''  =      -  ^'^  ******  *  " 
.^'  (sin'-'  i  11)      a'  sin  /<'  z=  -  :^"  A'.-  sin  /  « 

It  will  1..^  ol)s(M-vod  tlnit  all  the  coefficients  having  as  a  factor  c_itl.er  ^'  sin  I  n  or 

2  sin  i  n  cos  i  n  vanish.  ,     ,      1      ^ 

The  value  of  a  apparently  is  not  r.-a.lily  determined  dire.-tly  l.y  least  .scinares :  wo 
shall  therefore  assunu.  sevral  values  of  this  .p.antity.  an.l  as..-rtain  l.y  which  vahn.-  tlu, 
n.u.litions  can  l.est  i,..  satislied.     Tin-  lollowing  are  th.'  ahhreviate.l  values  ol  the  pur.'ly 


triiionometric  sunnnati.nis : 

sin  6i  n 
2  cos  f  II  =r    . — V —  =  c 
8111  2  n 

1%  sin  n  +  sin  13  «  _ 

2  cos-  in—    -  -    7 —  -^  1 

2  sill  n 

^-,   .  „  .  13  sin  »  — sin  13/;  _ 

2  sm*  in  —  -        . —  *i 

2  sin  n 

It  we  solve  the  preceding  equations,  and  put,  for  brevity, 

c 


C  - 


C\  = 


1 3  Ci  —  c- 
13C1  — c- 


c 13 

13  c, —  C'' 

the  resulting  expressions  for  the  unknown  cpiantitics  are: 

//  =        L\2hi—   C^'//..  cos  in 
«  sin /(  =         C2hi—C^ hi  cos  in 

a  cos  yu  zz 2  hi  sin  i  n 


k  —        C\2  ki  —  C2  ki  cos  (  n 
a'  cos  /i'  zz  —     C2  ki  +  C^  ki  cos  i  n 

a!  sin  fj!  —  —  -     '^  h  si"  ^  « 
«i 


»>»> 


'I'lii-  pciidil  (»r/Miml  It  lies  prolml)!}'  Ixifwn'ii  liriccii  iiixl  twenty  yearn,  whi(^li  would 
miikf  the  value  nl"  //,  or  tli(;iuiniml  motion  ol'  llie  iiiet|imlity.  lie  hetweeii  iS'^  and  24". 
Tlie  lollowiiii;  iire  tlie  values  ol'  the  various  t|iiaiitities  *le|>eii*liiig  on  n  tor  the  tlitVerent 
values  of  n  between  these  liniitri  : 


logf 

logfi 

log  S\ 

logf 

log  t' 

log  C, 

0.756 

0.715 

0.S93 

9.213 

9.17a 

9.571 

0.705 

0.707 

0.898 

9.097 

9.099 

9.506 

0.644 

0.705 

0.900 

8.977 

9.038 

9447 

0.577 

0.709 

0.897 

8.858 

8.9()0 

9. 395 

o.4<)8 

0.718 

0.891 

8.734 

8.954 

9.350 

0  406 

0.731 

0.88a 

8.604 

8.939 

9.31a 

0.391 

0.747 

0.870 

8.453 

8.()0<, 

9.276 

0.143 

0.765 

o.Ssf) 

8.275 

8.897 

9.346 

n 

S4|iin>M 

£  hi  cos  1  n 

S/t<slni» 

iki  cos  in 

a 

18 

+  11.48 

+     I .  '/> 

-     4.66 

-     4.66 

'9 

+  11.66 

+     1.52 

-     4. 08 

-     5.04 

ao 

+  ir.78 

4-     1 .09 

-     4  69 

-     5. -to 

ai 

+  11.83 

+     0  63 

—     4. 08 

-     5.73 

23 

+   11.81 

+     0.29 

-     4.00 

—     0 .  04 

33 

+  11.73 

—     o.oS 

-      4.02 

-     0.33 

34 

+  11.58 

-     0.44 

-      t.57 

-     0  60 

25 

1-  It. 37 

-     0.78 

-     4-5" 

-     6.80 

The  precediiisf  e(|natioiis  now  jrive  the  following  separate  values  of  the  unknown 
quantities,  eorresponding  to  tlu;  various  assumed  values  of  «: 


n 

h 

a 

f 

* 

.  a' 

f' 

0 

1, 

, 

„ 

. 

18 

0.73 

J.53 

164.0 

0.73 

l.Sl 

160.8 

19 

0.69 

1.53 

165.2 

0.61 

1. 71 

159.7 

20 

0.66 

1.53 

166.3 

0.49 

1.62 

158.5 

31 

0.63 

1.54 

167.3 

0.39 

'.53 

•  57.2 

22 

0.61 

>.55 

168.1 

0.31 

1.47 

156.0 

23 

0.60 

1.57 

i6q.o 

0.23 

1.42 

154.8 

24 

0.58 

'.59 

169.8 

0.17 

1.39 

'53.6 

25 

0.56 

1. 61 

170.4 

O.II 

1.36 

152.6 

There  can  he  litth;  serious  doubt  that  in  the  case  of  the  pnisent  inerpiality  the 
theoretical  values  of  /j.  and  /<'  should  he  the  .same;  and  it  is  also  |)rol)al)le  that  those  of 
a  and  a  may  Ix;  suhstantially  identical.  Th«!  small  ditrerenccs  between  the  values  of  a; 
and  a'  and  of  //   and   //'  add  so  much  weight  to  this  probability  that  we  shall  make 


2:^ 


icr  sdliition  of  Hie  (■(|iiiilions  on  tlic  siiiti»(isili(in  llmt  a'  —  ir  athl  /<'  —  /i.     'I' 


niiotl 

null  i'((iiuti()iiM  tliun  Ix'coiik; 


If  nnr- 


13  //  —   frtrsin  ft  =      ^/i^ 
—  ck-\-  iT,am\  /.i  z=  —  ^ hi 


cos  /  n 


-:>■/,■. 


1 3  A  -f-  ca  cos  ju  z=      ^  /{i 
c  A'  4-  1 3  a  cos  //  —      2  ki 
The  solution  ot  these  equations  is: 


cos  I 


n-:^h: 


<ni  /  n  —  N, 


sin  /  )i 


•3   — '  13'  —  '" 


//. 


'3  —  '  '3  — ' 

a  cos//—      .,  '^    .^«S.,  —      ,/      .,  :i'/-i 

A  conipiiris.Mi  of  tiu;  separate  solutions  ol  the  e(|iuilions  in  h  and  /-  shows  that  tlu! 
valui  .tf  n  which  best  satis(i(!s  the  (Mtuditions  lies  hetweeu  22''  auil  25^.  The  values 
of  /(,  /•,  a,  aiul  /i  were  therotlu-e  (hiriviid  only  from  the  last  etjuations  for  tin:  last  four 
values  of  ti.  For  each  of  th(!S(!  separate  valm-s  of  «,  tin;  eorrespondinir  valin-s  of  //,  au<l 
ki  \v(!r(!  C(  uiputed  from  the  formula' 

//,  =  /i  —  (r  sin  (/<  -f-  i  h) 
lii  zzk  -\-  a  cos  (/<  +  '  «) 

ii)  which,  it  will  be  rcniemhcred,  the   index  /'   is  simpl)-  the  nuniher  of  the  year  I'rom 
1868  ;   so  that  we  have, 

For  1862,    iz=i  —  6 
For  1863,    i  zz—  5 
etc.,  etc. 

These  computed  values  of  //;  and  /<■,  were  then  compared  with  (he  valiuis  derived 
directly  from  observations,  and  given  on  page  20,  and  the  sum  of  the  s(puires  of  the  out- 
standing residuals  was  taken.  Tlu;  valutas  of  the  unknown  (piantities,  together  with 
the  sum  of  the  squares  of  the  residuals,  are  as  t()llovv : 


22 

23 

25 


+  0.06  I  +  0.34  '    1.54 

+  0.63  1  +  0.27       1.52 

+  0.61  !  4-  0.20        I. 51 

+  0.58  -t-  0.14  I     1.49 


/' 

V 

0 

lf)1.2 

3.207 

161.3 

3- '70 

I6I.5 

3.246 

l6i.7 

.1.441 

The  sum  of  the  s({uares  becomes  a  minimum  for  n  zr  22^.8,  showing  a  j)eriod  ot 
the  inequality  of  15^.8,  witli  a  possible  error  of  a  year  or  more.  The  formula'  for  //<  and 
ki  thus  become : 

A.  =  +  o".64-  i".52siu(i6i°.2  +  22^.8/) 

A<  =  +  0".28+  l".52COs(l6l°.2  +  22°.8/) 


9A 


S-*-! 


^^ 


from  whifli  we  liavo  flio  following  comparison  of  tli(!  compiilwl  iiiid  ol)servc(l  values  ol 
hi  aiitl  l\: 


Year. 

h 

/•, 

C. 

0. 

O.-C. 

c. 

0. 

0. 

-C. 

1862 

+ 

O.OI 

+ 

0.04 

„ 
+  0.03 

+ 

1.67 

+ 

1.23 

_ 

0.44 

1863 

■  - 

0.48 

- 

0.6} 

0.16 

+ 

1.32 

+ 

t.:s 

+ 

0.46 

1864 

- 

0.79 



1.07 

—  0.28 

+ 

o.So 

-1- 

1 .09 

+ 

0.29 

1865 

- 

0.8S 

- 

1.03 

—  0.15 

+ 

0.22 

- 

0.15 

- 

0-37 

1866 

- 

0.74 

- 

0.47 

+  0.27 

- 

0.38 

+ 

O.IO 

+ 

0.48 

1867 

- 

0.37 

- 

0.93 

—  0.56 

- 

0.85 

- 

0.36 

+ 

0.49 

1868 

+ 

0.14 

+ 

0.34 

+  0.20 

- 

I.  If) 

- 

1.46 

- 

0.30 

1869 

+ 

0.74 

+ 

1.67 

+  0.93 

- 

1.23 

- 

r.5f. 

- 

0.33 

1870 

+ 

1-33 

+ 

1.4S 

4-   0.15 

- 

1.07 

1. 14 

- 

0.07 

1871 

+ 

1.80 

+ 

1.65 

-   0..5 

- 

0.70 

- 

0.36 

+ 

0-34  i 

1S72 

+ 

2.09 

+ 

2.15 

+   0.06 

- 

o.iS 

0. 12 

+ 

0.06 

1873 

+ 

2.15 

+ 

1. 91 

-   0.24 

+ 

0.42 

^- 

0. 16 

- 

0.26 

1874 

+ 

1.98 

+ 

1.92 

—   0.06 

+ 

1. 00 

+ 

0.60 

0.40 

Tiio  ]frol»al)le  residual  for  each  year  is  d' .2']. 

We  have  supposed  tlie  hypothetical  inequality  of  longitude  to  Ik;  of  the  form 

idv  zz  lii  sin  g  +  ^'i  f^**'"*  -  ■ 
Sub.stituting  in  this  the  periodic  part  of  //(and  /,,  and  replacing  /  by  /,  which  now  reprc- 
sent.s  the  time  in  years  from  186S.5,  it  becomes: 

^vzz  i".52  sin  (g'+25i°.2  4-  22^.8/) 


or 


Jv=  i".52  sin  [5-+  22°.H  (  r-  IS57-5)] 


The  entirely  unexpected  character  of  tlu;  periodic  term  thus  brought  to  light  ren- 
ders its  verification  by  a  longer  series  of  observations  very  desirable.  For  tliis  purpose, 
we  need  comparisons  (tf  ol)servati()i.s  previous  to  1862  with  Hansen's  tables,  because 
none  of  the  older  tables  with  which  comparisons  have  been  made  are  accurate  enough 
for  the  purpose.  Now,  the  Greenwicii  Observations  for  1859  contain,  as  an  appendi.x,  a 
Cf»mpari.<on  (»i  tlu;  longitudes  and  latitudes  from  Hansen's  tal)les  with  Greenwicii  ob.serva- 
tions  from  1847  to  1858  inclusive;  ;  and  I  have;  utilizeil  the  comparison  of  the  longitudes 
derived  frciiii  meridian  observations  in  the  Ibllowing  way  : 

A  list  of  limiting  dates  t  tenths  of  a  day  was  made  out,  including  the  whole  twelve 
years,  and  sliowing  lietween  what  dates  the  moon's  m(;an  anomaly  was  found  in  each 
sextant.  Tiie  sum  of  the  errors  in  longitude  given  liy  the  meridian  o'oservations  was 
then  taken  during  the  period  that  the  anomaly  was  found  in  each  sextant.  None  of  the 
corrections  found  in  the  first  part  of  this  discussion  were  aj  olied,  for  the  reason  that 
mo.st  of  them  could  l)e  treated  as  accidcnital  errors,  and  the  means  could  be  taken  so  as 
nearly  to  eliminate  the  effects  of  tlie  larger  ones.  A  specimen  of  the  form  chosen  is 
here  given.      Under  (;ach  of  tlie  several  values  of  g,  given  at  the  tops  of  the  several 


, 


25 


re- 


oulmiiiis,  is  .shown,  firstly,  liif  dale  al  wliic.li  a  liail  thai  particiihii-  vahic:  and,  scroiidly, 
the  sum  of  the  residuals  in  h»niritu(h'  dnriuir  tlic  period  of  4''.6  Itctwct'ii  that  dalo  and 
the  one  iiiixt  lidlowing,  logetli(3r  witli  the;  inind)cr  of  tlic  i-jsiduals,  tlic  hitter  iieinix  in 
small  sidiserijit  tiirures. 


I. in. 

V<1,. 

M:ir. 

.\|.ul 

May 

(line 

[illy 

Auk. 

'  Auk. 

j  Sepl. 

;  Oct. 

I  Nov. 
Dec. 


Jan. 
Fcl.. 
Mar. 

April 
May 
May 
I  u  ru- 
in ly 
Auk. 
Sc-|.t. 
f)<-t. 
Ndv. 
Dec. 


=  o   + 

I").li-     2.1)1 
Id.  I  -     1  .(), 

>5.7        .    .    ; 

'2-3  .     .      ! 

<).S       .    . 

(>.2  f     2..S, 

"■4-  II. 3j: 

28.0-t-     5.IJ;,   ; 

24.fH-l2.2:i  ! 

i 

22.1  t  I2.2< 

l')-7-    I.2i 
16.  I-    3.4.J  , 

I 
I 

1848.  I 

12-7-   7-3-' 
9-3-   S.I, 
7.0-    r.."*,  ' 
1.5        .    • 
2.0 
2c)  ,6 

2^ .  2 

23.7  .     . 

20.3+    1.2, 

I'l.9^  22.5, 

I 

14.4+  5-ii  ; 

I  1  .  o  -t-    f) .  ()n 

S.5-  (i.sJ 


,<:  -■  '■')   f 


Ian. 

I'd,. 

Mar. 

April 

M.ay 

)unr 

July 

Auk. 

Sept. 

Sept. 

Oct. 

N.iv. 

I)i-c. 


Jan. 
Mar. 

.'UMil 

May 

iiinc 

[line 

July 

Auk. 

Sept. 

Ort. 

Nov. 

•)cc. 


IS47.    ' 

2.(.2-    3, 

:i..74  „, 

20.3-  3 

U<.<)  . 
14.4 
in.S 

S.4  . 

5.')  . 

I.'■|^  3 

29.2  . 

2f..7+    fi 

23-3 ^    0 
20.7       . 

1S4S. 

173       . 
13.9-    fi, 

12.5-  4 
'  I .  I  —  4 , 
fi.fi-   7 

3-2-  8 
30.8-   2 
28.3 
24.  ij 
21.5 
l').o 
15.6+12 
13.1+  5 


.1,'=  120    4- 


Jan. 
Jan 
Keh. 
Mar. 
April 
May 
June 
July 

A  UK- 
Sept. 
Oct 
Oct. 
Nov. 
Dec. 


Jan. 
Feb. 
Mar. 

April 
.May 
Inne 
July 

•  I  Aug 
■  I  AUR. 
.    ^  Sept. 

.    I  Oct. 
8:,  I  Nov. 

li    Dec. 


1847.  " 

1 . 2  I  I . 

2.S.S  I  3. 

25-3+  4 

24   ,)  f  o. 

21.5'  3- 

19.0+  2. 

15.4-  I. 
13.0 

c,.(,  . 

f).2 

3-8  ■ 

3t.3+  I- 
27.64-11. 

25.3+  o. 
1848. 
2r.9 

15. 5-  I. 

I7.H-  4. 

13.7-  I- 

II  .2+  I. 

7.8-  o. 

5.4—  o. 

l-O-  5- 
29.5 

26 . 1 
23.6 

20 . 2 
17-7+  7- 


3:     Ian. 

f.   r.  I.. 

::      M.r. 

1,  Mar. 

2,  .April 
8..     May 
4,      I  line 

■  J'll.v 
.  I  A  UK. 
Se|)t. 
.  !  Oct. 
ri,  .Nov. 
4:1  Dec. 
7i     r)ec. 

i  , 
.    i  Jail. 

4.  '  Feb. 

-:.  '  Mar. 

5;     .\pril 

n,     M:>y 

9,  I  June 

f.,     July 

4.     Auk. 

Sept. 
.  iSept. 
.      Oct. 

Nov. 
O;    Dec. 


=  181/  4- 

1847. 

2 .  M 

1.9.- 
29.5  - 
26.0  f 
23.6+- 
20 .  o  — 
17.6- 
14.2 

10. 8 
8.4 
4  9 
2.2 

29.9  + 
I84S. 
26.54- 
23.1- 
21  .  7 

18.3- 

1 5  •  8  -t- 
12.  1- 
10. o  — 

6.5- 

3"-7- 
28.2 
24.  S 
22.3 


3-7i 
3-7. 
0.4, 
2.71 

3  ■  ',:i 
O.6.. 
1.6, 


"■7> 

0.2, 
I.41 

I  ■  -I 
1.4: 
o .  2 . 

4  •  "■  : 

3->i 
6.7.- 
1 .0. 


Ian. 

I'.b. 

M:ii. 

.\piil 

.\lJiil 

May 

June 

July 

Auk. 

Sept. 

Oct. 

Nov. 

Dec. 

Dec. 

Jan. 

Feb. 

.Mil 

.\pril 

.May 

I  line 

July 

Auk. 

.Sl;.t. 

0,1. 
Nov. 
Nov. 
Dec. 


240    -t- 

1.847.      ' 
10.4  t-   2 

7"+  5 
(1.5  »-  2 
3.1 

30.(1 

28.2—  I). 

21.6-    2 


b,      Feb. 

■3,:  Ma'. 
Apiil 
.       May 
3.'.     J  line 
1 1      June 


3.61 


I-  2.9,  ;  July 
AiiK. 
Sept. 
Oct. 
Nov. 
Dec. 
Dec. 


1S.8 
15.4 
13.1) 

'J- 5        • 

6.S 

31-5 
1S4S. 
31. I 
27.7-  n 

2').  3 

22.9  t     2 

20.4+     9 

17.0 

14.6(10 

I  I  .  I 
7-7-   5 
5-3+   2 
1.8  f    I 

29.4-   9 

26 . 9 


'  Feb. 

I 

;  Mar. 

Mil. 

A|iril 
!  May 
Tune 
July 

.       AiiK. 

i:i     Sept. 

O:,       Oct. 

9..  '  Nov. 
4:i  Dec. 
.      Dec. 


4i 


300  t- 

1847.       " 

■  ■i-o  .     . 

-b  ..! 

II. I  .  .  i 

7.7  t-  3 -21 

5.2-1-  I.O, 

I.S-t-  4.1,  I 

29.2-)-  1.9, 

2b.  8  t-  6.8:, 

23.4+  S-ljj 

20.04-  I  Ah 

17-6-  7-3.' 

14. 1  o.o,  [ 

11.4  ~  1. 6., 

3')-i- 
1848. 

4-7 

3  3 
3"') 
27-5 


2.8 


Ml 
O.Ij 


21.  b  t- 

19.2- 

15.7  )-i7-4i 

12.3  +  15.7:1 
9.8+  8.3, 
6.4-  4-9:1 
4.1:-    7-2:i 

31-5-   5.71 


If  we  tollow  any  one  of  these  vertical  columns,  we  shall  liiid  that  the  dates  corre- 
sjtmid  sueeessively  to  all  points  of  the  lunation  in  a  |»eriod  of  .|I2  days.  Tin;  first 
(d)servations  of  each  period  will  he  Iho  last  ones  of  the  lunation,  and  the  last  ones  those 
made  immediately  aftur  nt^w  moon.  Between  <'a(di  pair  <il'  periods  will  lie  a  gap,  gen- 
erally of  three  or  lour  months,  during  which  the  moon  was,  at  the  corresponding  points 
of  mean  anomaly,  too  near  the  snii  to  he  ol»served.  li  the  t'Wservations  are  etpially 
scattered  Ihrongh  each  period,  all  the  errors  arisinir  troni  erroneous  senu-dianietor  and 
|)arallactic  inetpialily  will  he  eliminated.  Tin;  <reneral  minnteiiess  of  these  errors,  and 
their  approatdi  to  a  balance  during  each  id' the  periods  in  (piestion,  are  such  as  to  render 
them  insignificant,  if  we  takt;  the  m";in  results,  not  hy  years,  Iml  by  periods.  This  is 
the  course  adopted;  the  partial  periods  al  the  iieirinninLr  iiiid  end  of  the  entire  scries  of 
oh.servations  being  omitted.  The  first  period  actually  employed  was  that  corresponding 
4  m 


ii 


Ismimm 


20 


\i 


m 


to  tlic,  soxlaiit  240  -300 ',  ill  wliicii  llic,  first  ohscrvation  was  made  011  Jaiiiiuiy  10, 
1847,  and  tlio  last  on  Soj)tfinl>or  18  of  (lie  same  year.  Tiu*  last  jxirioil  coricspomlcd 
to  the  sextant  180-240'^',  the  last  observation  in  which  was  on  November  13,  1858. 
There  were,  in  all,  ten  periods  corresponding  to  each  sextant,  and  hence  ten  sets  ol 
(;(|uatioiis,  each  liiving  iiu^an  values  of  //,  /i,  and  SI  for  periods  extending  throuiili  a  little 
more  than  a  year.  Each  residual  gave  an  e(piation  of  con<lition,  for  th<!  eoellicienls  of 
whi(di  th(!  mean  value  corresponding  to  tiie  entire  sextant  was  taken.  Tiiese  values  for 
tiie  sev(!ral  sextants  areas  follow: 


1 

^ 

sin.f 

COS.!,' 

sin'i' 

sin^cos,f 

COS«A' 

1 

0-    60 

-(-  0.4S 

+  0.S3 

0.23 

+  0.40 

n.69 

2 

60-  120 

+    O.rjC 

0.00 

0.91 

0.00 

0.0c 

3 

120-  180 

+    0.4S 

-  0.83 

0.23 

-  0.40 

0.69 

^ 

180-240 

-   0.4S 

—  0.S3 

0.23 

+  0.40 

0.69 

5 

240  -  300 

-   r-../. 

0.00 

0.1)1 

0.00 

(J.  00 

6 

1 

300  -  360 

—  0.4S 

+  0.83 

0.23 

—  0.40 

0.69 

The  sums  of  the  residual   errors,  corresponding  to  each  period  and   each   sextant 
arranged  in  chroiudctgical  order,  togetiicr  with  the  number  ol"  residuals  of  which  each 


sum  is  formed,  are  as  follow 


Mean 
(late. 

/  =  5 

1  =  6 

»  =  I 

(=2 

! 

=  3 

»  =  4 

1847.8 

+     6.5 

-f    14. -1*1 

+    IS. An 

n 

+ 

0 .  Oji 

-   if>.7i7 

1848,9 

■      6.7 

+       8  .7 

-  33-f^i: 

-     1.931 

-i 

23.2i» 

■f-  31-517 

1S50.1 

f      1.5 

-   34I17 

—  40  ■9*1 

—     9.1^1, 

+ 

22 .  2m 

+  33. 9» 

.85.- 

-       4  ■  5  :l 

-     5')-Ai-: 

-  50. 7i'., 

-  23.5.^1 

- 

4.821 

+    20.6...,, 

IS52.4 

-  42.8.J 

-    50.O..;l 

—   48.0ir, 

-  21.51S 

+ 

35  .Oil' 

+  25.4.;, 

IS53.5 

-    3I-2JJ 

—  106.9.., 

-  f)3.f'ii 

+       I.2„ 

+ 

6.0.^1 

-  38.>>*> 

1854.6 

-   3"-3i; 

-   94  •'^';i 

-  35  •4« 

+      4  •  2w 

+ 

i.7h 

-  24.48" 

1S55.8 

-    24-3n 

—     30. 0|,. 

-     7  •3:.. 

-      6.9,,, 

- 

22.81s 

-  41. Oj., 

1856.9 

-    3f'.2 

-   23-8,. 

+   «5-4i. 

+     4.2jf. 

- 

48.511 

-  77.017 

1858.1 

-   51  •■>:•. 

-  48. 9.:, 

-    5f'.7ni 

-  47 -Si'.' 

— 

7&.9« 

-  46.2i„ 

The  dales  given  in  the  left-hand  column  an^  those  corresponding  to  the  mean  of 
each  liori/.ontal  line. 

Piil'tiiig  Sj  Ibr  I  lie  nieaii  value  of  sin  ij[  corresponding  to  the  index  i,  as  already 
given;  r,  for  that  of  cos  ir ;  and  W;  for  the  corresponding  numi>cr  of  oltservatimis,  tiie 
iiornial  etpialions  are: 

// ,.  J/+  {^  V ,  .V,)      //  -I-  (2  Hi  c,)     k  =  ^^  i\ 
(V  „,.  .v,.)  Jl  +  ^^  „ .  ,.;^)    /,  +  (^' ,, .  ,s.. ,:.)  /,  -  2  .V,  r, 
(V  /^.  r,)  Jl  -f  {2  »,  s,  ci)  I,  -f-  {:>:  ;/,  c-)    h  -  2  r.  r, 

The  values  of  Ii  and  k  thus  given  by  the  normal  cfpintions  formed  from  the  system 
ol  residuals  siiown  in  eac.ii  iiorizoiital  liin-  are  shown  in  tin!  next  tabh\  wliich  also  shows 


27 


tlu!  way  ill  wlii(;li  tlifv  arc  tn-alfd.  For  tin;  saUt;  of  C()iii|tlt'lt'ii('ss,  llic  corrfsiioinliiifi 
<|iiaiiti('n's  alreaily  toiiiKl  for  tlx-  jx^'riod  1S62-74  an;  added,  and  iiRdiidcul  in  tlie  discus- 
sion, wliicii  now  procet'ds  a.s  follows;  tin;  nn;lliod  adophnl  l»i'in<^  one  wliicli,  lliouirli  less 
rijjorons  than  the  former  one,  will  show  in  a  stronger  liulil  the  evidence  on  which  the 
new  itieqnality  depends. 

As  the  basis  of  the  discussion,  we  take  the  indepenilenl  values  of  //  and  /',  derived 
from  each  series  of  (d)servations,  which  values  are  iriven  in  the  second  and  third  columns 
of  the  tal)le.  A  preliminary  comparison  of  the  first  series  of  values  ( 1S47-58)  with 
the  values  of  //  and  k  derived  from  the  formuke  already  ifiven  indicates  a  din)inution  of 
th(!  constant  terms  of  those  (piantities,  so  that,  insl<;ad  of  +o".64   and  +  o".2.S,  they 

I)econie,  as  a  first  approxim  .tion, 

//«:r:+o".50 

A-or=  +  o".io 

These  constants  an-  now  subtracted  from  the  values  ol  h  and  /r,  leavini,'  a  series  of 
residuals  <fiven  in  the  fourtli  and  fifth  eolunnis,  which,  if  the  peiimiic  leini  under  in- 
vestigation has  no  existence,  shouhl  I»e  regardeil  as  due  to  ernirs  of  i)l)servation,  and,  in 
the  contrary  case,  slutuhl  be  representablc  by  the  formula' 

h'  z=.  —  a  sin  (yu  -f  nl^  -f-  accidental  eriors 
k  zn       nr  cos  (//-(- w/) -f- accidental  errors 

To  show  clearly  how  far  they  are  thus  re|tresenteil,  we  deteiniine  a  coeflicient,  ix, 
and  an  anixle,  A',  i»y  the  (;(juations 

(X  sin  A'n  —  h' 
acos.V::^       /r' 

Tin-  next  two  columns  ifiv(!  the  several  values  of  a  and  A'  tiius  ol)tained.  'I'he 
nearly  rciiular  proi^ression  of  the  angle  iV  is  too  striking  to  l)e  overlooked.  To  st.-e  how 
nearly  this  angle  can  be  represented  as  one  incrtjasing  uniforndy  with  tin,'  time,  \ve  .solve 
the  iH'CCSsary  eipiations  of  condition  by  least  sijuares.  Il  is  ol)vious  that  tin;  greater  the 
value  of  a  the  more  certain  will  l)e  the  value  of  N\  we  tluM-efore  <rive  weights  propor- 
tional to  a.  Moreover,  weights  nearly  twice,  as  great  in  proportion  ar<' iriviMi  to  tin; 
.second  .series  (1862-74)  as  containing  the  results  from  two  observatories,  and  beinii 
more  carefully  corrected.  The  values  of  /<  and  n  thus  obtained  l)y  the  method  of  least 
s{[uares  are  : 

//  —  i64'.6±4   -4 
H  —     20  .S  ±  o  .47 


Si 


m 


'•A 

\ 

an, 


'31 


1:^1 1 


The  pndiable  error  of  a  value  of  .\' (d'  weight   unity  conies  uiit 

'i'he  residuals  still  uutstandinif  are  shown  in  the  coliiinii  JS.  This  valui'  of  //  is 
2""  le.ss  than  that  found  from  the  secontl  seri(!s  of  (dtscrvations  alone,  and  an  examination 
(d'tiie  residuals  shows  that  there  is  a  real  discordaiu^e  !)etween  the  values  (d'the  anu'ular 
motion  of  .V  uiven  liy  the  two  series.     It  is((uit(!  liki-ly  Ihat  tiie  relative  weights  a.«;signed 


HP 


ij 


iii 


28 

to  tlic  (tiller  scrii's   of  ohseiviitiDiis  an-  twice  as   great  as  tliev  slnmlfl  l)e,  and  that  tlie 
most  [nobalilu  value  (»!'  tliu  angle  Allies  nearly  halt-way  Ijetwecn  lh(!  two  values 

l6r\2  +  22'\^{t—  1868.5) 


and 


i64''.6  +  20^.8  (/—  1868.5) 


tijund   from   the   last  series  alone,  and   Iroin  the  two  conihined.     I  judge  that  I  he  most 
prohaljle  value  is 

iV=  i63'^.2  +  21^.6  (/-- 1868.5), 

and  that  the   proltaltle  error  of  the   aiinuu'   motion   is  moni  than  hall'  a  degree,  hut  less 
than  a  degr(  e.     The  eohuiiii  _/'.V  shows  the  residuals  ifivon  hv  this  value  ol"  N. 


iMc;in 
date. 


1847.8  i 

1543.9  I 
1850. I  ] 
1851.2 
1852.4 

«853.5 
1854.6 
1855.8 
1S5&  (J 
1S5S.1 

I  1862.5 
1863.5 
1864.5 

♦1865.5 
'  1 866. 5 
.867.5 
1S6S.5 
l86c).5 
1S70.5 
1S71.5 
•S72.5 
1S73.5 
1S74.5 


-  0.08 

-  0.55 

-  0.20 

-  U.32 
+  0.26 
+   1. 10 

+  '.45 

4-  0.77 

f  1 .  76 

-  0.17 

+  0.04 

-  0.64 

-  1.07 

-  1.03 

-  0-47 

-  "03 
y  0-34 
+  1 .  67 
+  1.48 
f  l.f.5 
1-2.15 
I-  I  1)1 
+  i.y2 


+  <).55 

-  1. 38 

-  !.<)! 

-  I  .  1)2 

-  2.45 

-  I. S3 

-  1.40 

+   i>.3i 

+    1 .  82 

I 
4-  11.66  I 

i 
-f   1.23 

+    1. 78 

¥    I.oi) 

-  1J.15 

+     O.  ID 

-  '>-3^i 

-  I.4''' 

-  1.5'' 

-  1.14 

-  0.3^ 
-  o.  12 
1^  0.16 
H   0.60 


-  0.5S 

-  1.05 

-  o .  70 

-  0.82 

-  0.24 
4-  0.60 
+  "-OS   j 
+  "-27  I 
'h  1.26  ! 

-  0.67  ! 

-  0.46  I 

-  I. 14  I 

-  1-57   i 

-  1-53  ! 

-  i>.<)7  j 

-  1-43 


0.16 


)-  1. 17  I 

+  o..,S   I 

I  1. 15 

h  1.65    : 

I  1.41 

(  1.42 


+    0.45 

—  I.4S 

—  2.01    i 

I 

—  2.02 

—  2-55  ^ 

—  1.98  I 

—  1.50  j 
+  0.21   I 

+  1-72   { 

+  0,50  I 
! 
+  1.13 

+    1.63 

+  <'-'W  j 

—  0.25 
0.00 

—  0.46 

—  1.56 

—  1.66 

—  1.24 

—  0.46 

—  0.22 
+  0.06 
+    0.50 


a 

1 

0 

0.74 

52 

1.82 

145 

2.13 

i6l 

2.13 

15S 

2.56 

175 

2.07 

>97 

1.77 

212 

0.34 

30S 

2.13 

328 

O.S3 

50 

1 .22 

22 

2.03 

34 

1.35 

58  ! 

».55 

99 

0.97 

<JO 

1.50 

108 

1-57 

174 

2.03 

215 

1.58 

2IS 

1.24 

248 

1.66 

262 

1. 41 

272 

1.50 

289 

I 

3 
3 
3 
4 
3 
3 
h 

3 
t 

3 

5 
5 
4 
2 

4  I 
4! 

SJ 
5 
3 
I 

:! 


/I  +  Itt  AN 


94 
118 
141 
165 
IS.) 
212 
236 
260 
284 
3<J7 

40 
61 
81 
102 
123 
144 
165 
1S5 
206 
227 
24S 
269 
289 


+  42 

-  27 

-  20 
+  7 
+  14  I 
+  15 
+  24 

-  4» 

-  44 

-  I1J3 

+  18  i 

+  27' 

+  23 

+  3  , 

+  33  I 

+  3(> 

-  9 


A 

N 

4 

24 

44 

36 

- 

H 

u 

+ 

2 

-f 

12 

-  (<t> 

-  55 

-  112 


12 
21 

19 
1 

3" 
34 
II 


:  -  3u 

-   30  ; 

1  —  12 

-  12  t 

—  21 

—  21) 

-  14 

—  12 

-  3 

—   I 

" 

^   -i 

'I'he  old   and    new  series  ol   ohservalions  agree  well    in   giving  lor  the    value  ol   the 
eoeHieient  ol'  lliis  li'rni. 

'I'iie    old    series,  rr  —  i".66 

'{"he  new  series,  n  r:z  l".55 

The  (died  ol'  the  accidental  errors  will   lie,  on  the  whole,  to  increase  the  value  ol' 
the  coellicieMf.      I  consider  thereliire  that  tht^  value 


<f  —  I    .50 


29 


iiiiiy  !)(■  iiiliiptrd  iis  llu!  luosl  prohabli!  wliicli  ciiii  l>c  derivotl  from  all  llu;  ubsorvatioiis. 

11'  we  siilitiiicl,  IVoui   fiicli  value  of  //  aiul  k  in  the  preceding  table,  tlio  perindic 
portions 

//zr-  i".50>iH  [163". 2  +  21^.6(^-1868.5)] 
/,'=       i".50(:os  [163^.2  +  2r'.6(^- 1868.5)] 
\\\n\  liikc  llii'   iiu'iui  viilue  of  llie  outstanding  remainder  for  (!uch  scries  of  oltservalions 
we  lind  it  to  Ix;  as  follows : 

Old  scries,  //o  =  +  o".33;    /-o  =  — o".i7 

New  scries,  h^  —  -\-  o".65  ;    A-q  =  +  o".36 

Till!  (liUcnMici's,  o".oi  and  o".o8,  between  these  last  values  and  tliost;  found  on  page 

23  arise  from  the  dillcrcnl  value  of  the  periodic  term.     I  consider  that  the  results  of  the 

second  sciics  arc  entitled  to  three  times  the  weight  of  thonc  of  the  first,  and  shall  there- 

tiirc  put  l()r  the  dctinitive  values  of  h  and  k, 

//=z  +  o".57  +  /'' 
Z— +  o".23  +  A' 

The  corresponding  corrections  to  the  eccentricity  and  longitude  of  perigee  arc: 

Sf-  —  0".2()  X 

r67r  —  -\-o".\2  / 

<5;r  — +2".2 

Tlie  corrcclions  to  the  moon's  longitude  are:  / 

'V  n  —  // sin  ij'  —  /-cosi'  / 

r:  — o".57  sin  i,'  —  o".23  cosir+  i".50sin  {g  +  N  —c)o'^). 
Tlif  last  term  istht;  liithcrto-uiisus[)ected  ineipiality  indicated  by  observations,  but  not 
vet  known  to  be  given  by  theory.      It  may  be  either  an  inequality  of  the  ecointricity  and 
perigee  having  a  period  cd'  about  \(i%  years,  or  one  of  the  moon's  mean  longitude  having 

a  period  of 

2  7''.4304  ±  o''.004o 

Substituting  first  fi)r  A',  and  then  fori,',  their  values  in  terms  of  the  time,  the  expres- 
sion fi)r  the  inciinalitv  of  longitude  becomes 

I  w  <  * 

i".50  sin  [-  +  73^2  +  2i'^.6  (/  -  1868.5)]  =  1  ".50 sin  (56^.8  +  13^.12413  0, 

7- being  tiie  tinn;  in  days  (;ounted  from  fjrceinvich  mean  noon  of  1H50,  Jan.  o. 

It,  nould  pcriiaps  l)t!  premature  to  introduce  so  purely  (unpirical  a  term  as  this 
into  lunar  tables  for  p<'rmanent  use;  but  where,  as  at  present,  it  is  recptisite  to  obtain  the 
cnrrcelioMs  In  the  tal)l('s  (hiring  a  limited  period  with  all  possible  accuracy,  tht;  eviih'nce 
in  IJivor  of  the  it  ality  of  th(!  term  seems  strong  enough  to  justify  its  introduction.  The 
niilv  .ippiirciil  cause  to  wliicii  the  term  can  i)e  attributed  is  the  attraction  cd"  sonic  one 
of  llic  planets. 

In  the  investinalioii  (d'  corrections  to  the  longitu<le,  it  only  remains  to  determine 
tlie  slowly-vaiyiiig  conci'tions  to  the  mean  longitude,  or  to  «  f5r,  given  by  the  observa- 
tions. To  defeiiiiine  tlu'  errors  of  short  period,  we  have  applied  several  corrections  to 
I  lie  residuals,  not  as  real,  but    only  to  render  the  various  ol)servations  comparable.     We 


80 

liavc  ii«»\v  to  con.sidur  (lit;  jMin;  results  of  obscivatiuiis  as  lliey  would  liavi;  l>i!eii  liad  tlicso 
corioctioiis  not  bt;cu  applied.  Tliesc  lor  tlio  second  series  of  observalioiis  are  loiirid 
I »y  taking  tin;  sum  01(1)  the  incjan  of  tlie  small  corrections,  ap[)!ied  on  account  of 
(>')S(!rvatjry  and  limb,  to  compensate  lor  the  systematic  dillerenees  l)etvve('U  results  from 
dillerent  lind>s  or  dilferent  observatories;  (2)  general  corrections  to  make  the  residuals 
in  the  mean  very  small ;  (3)  remaining  outstanding  corrciction  found  by  solving  the 
cipiations  of  condition. 

The  corrections  from  l)oth  series  are  as  follow:  the  corrections  sinc(^  1862  may 
be  very  closely  represented  by  a  term  increasing  uniformly  with  the  tiuK!,  as  is  shown 
l)y  the  last  two  tolumus. 

First  scries. 


Date. 

u>h 

Dalu. 

IS53-5 

II, h 

18.7.8 

-  0.15 

•• 
+    1-77 

1S48..J 

-  0.43 

iS54,f, 

+    > • 40  1 

1850.1 

+  0.32 

1855.8 

+   1-24  1 

1S51.2 

+   1. 13 

iS50.() 

+   1.50 

1852.4 

+  093 

185S.1 

+  2.40 

.■.;ti 


Second  series. 


Year. 

(I) 

+  <'-45 

(2) 
+    2.10 

(3) 
t    0.04 

II  ih 

a-\-l>l 



+ 

A 
1.07 

1S62.5 

+    2.5'J 

+   1.52 

1S63.5 

+   0.45 

+    1.2l> 

—  0.27 

T-      I. 38 

+  0.60 

+ 

0.78 

1.S64.5 

0.00 

0.00 

—  ".49 

-  0.4IJ 

—  0.32 

- 

0.17 

1S65.5 

-  0.15 

-    ■•>5 

-  0.62 

—    I  .  1J2 

-    1-24 

- 

0.68 

1866.5 

-  0.15 

—   2.m) 

-  0-75 

—    2.  (JO 

—  2.16 

- 

0.74 

1S67.5 

-  0.15 

-  3-40 

—  0.41 

-   3')6 

—   3.oS 

- 

0.88 

1 868. 5 

-  0.15 

-  4.05 

—  0.20 

-  4.40 

—   4.00 

- 

0.40 

lS6(j.5 

+  o.oS 

-  4.85 

—    0.2I 

-   4.')8 

-   4-92 

- 

0.06 

1S70.5 

+  0.08 

-  5.50 

—  o.o() 

-   5-51 

-   5.84 

+ 

0.33 

1871. 5 

0.00 

-  f'-35 

-  0.52 

-  6.87 

-   6.76 

- 

0.  II 

1872.5 

—  0. 15 

-  7.25 

—    0.22 

—   7.62 

-   7.68 

+ 

0.06 

1873.5 

o.ou 

—  8.30 

+    O.IO 

-  8.20 

-   8.60 

4- 

0.40 

1S74.5 

0.00 

-  9-45 

+    0.38 

-  9-07 

-  9- 52 

4- 

0.45 

§2. 
INVKSTICAIIOV  OF  THE   TOLAR   DlSTAIvICK  AN'D  l.ATirrDK. 

It,  is  a  siiiLnilar  «-irrii!ii.staiic('  lliat  diiriiig  the  lust  six  years,  at  least,  the  (li)si'rva. 
linns  of  tlic  niooa's  |ii»lar  <listaiu'(!  are  imicli  lt>ss  accurate  than  those  di'  its  riulit  ascen- 
sion. Wlietlier  this  is  to  he  altrilinted  to  lh»i  instrnnients,  or  wlietiier  it  is  a  resiiH  ot' 
f^rent  irreirnlarities  in  the  oiitrnie  of  the;  lunar  glol)e  in  tiie  polar  reitions,  cannot  at  pres- 
(Mil  i)e  (lecitled.  To  whatever  cause  wo  atlrihutt;  the  errors,  tiieir  existence  renders  a 
rit'orous  treatment  ot  the  in;lividual  observations  of  little  value.  We  shall  therefore, 
from  the  wiiole  of  the  errors  in  dediniition,  sook  to  olttain  the  i>est  corrections  to  the 
inclination  and  node  of  the  moon's  orl»it. 

Fvom  the  derivatives  of  the  moon's  d(;clinatiou  relatively  to  its  true  lonizitinle,  the 
inclination,  and  the  node,  whieii  have  already  boon  given,  wc  oi)tain: 

ot  4- 
dl     ^  do 


„ »      no  5,     no  „^   ,  (Id  „ 

no  zr        ''' +        o"  +    ,.  "' 


dS  .. 

di 


ly  heing   known   tVoni   tin;  data  already  givcMi,  the   (Hjuations  ol  condition  will   be 

thrown  into  the  li>rm 

d''^   ■  ^r^   .   dS  ^.        5.  ~       dS  f, 

r     ,      I  OO  -\-  -—  01    —  00   —     ,,    0/ 

I  do  '    di  di 

Vrom  llie  numerical  cxpressitnis  already  givcin,  we  have 

-'''^  <S/  —  sec  <^  [(0.40  +  o.oS  cos  0)  cos  /+  o.oS  sin  0  sin  /]  rV 
dl 

If  we  put 

fSA  =:tlie  (•orre<:ti(Hi  to  the  moon's  mean  longitude, 
K  ziz  0.40  -\-  0.08  cos  0, 
II  =:0.oS  sin  0, 

wo  iiave  tlie  quantities  of  the  first  order,  with  respect  to  tiio  eccentricities, 
—  [A'cos  /  -f  //"sin  /]  [i  +2  r  cos  (A  —  ;r)  ]  see  rS 

The  largest  terms  in  sec  S  are 

1.040  -f-  .016  cos  ^  —  .040  COS  2  A  —  .016  cos  (2  A  —  0), 
while,  it"  we  replace  /  by  the  mean  longitude.  A,  we  shall  have: 

/=r  A  -|-  2  c  sin  (A—  tt) 
sin  /  =:  sin  A  +  ''  ^^iu  (2  A  —  w)  —  v  sin  tt 
cos  /  zr  cos  X  -{-  f  cos  (2  A  —  tt)  —  t'  eos  tt 

I  s 

If  we  sid>stitn(»!  these  various  (luantitie.s  in  the  expression  tor  ,,  6/    \\(>   shall  lind 

tlf 


mmmum 


iPf*'i 


M- 


■MMH 


3^ 

no  sciisihlo  tciins  (Icpciidingon  tlio  sine  or  cosine  olllio  argmncnl  of  liililiulc,  A  —  0. 
we  siihslitiile  lor  SI  its  vulut!  in  <5A,  wo  sliiill  iind  tlu;  priiKiijial  Icriiis  in  cos  (!• 


,//  </\ 


to 


1)0 


A' COS  A  +  /Tsin  A  +  3  «  /vcos  (2  A  —  ;r)  +  3  c  //sin  ( :;  A  —  /t) 


-f  0.9  cos  /  —  o.:  sin  / 


III  consoqucnco  of  the  great  number  of  revolutions  ol'  \\u\  moon  tlirougli  wiiicli  tin; 
ol»s(!rvations  now  umler  discussion  extend,  T  have  considered  tlml  all  (!xcc|)t  the  first  two 
l(!rms  might  he  treated  as  accidental  errors,  which  would  cancel  each  oilier  during  the 
course  ol'  tlu;  oliservatious.  Using  for  S\  the  iiutan  correclions  lo  liie  moon's  loniiididc, 
we  have  the  lollowiug  values  of  the  correction  to  the  dcclinalion  for  Hiosi;  errors  of 
h>iigitude: 

Year,  CoiTi'ction. 

1862. 
1S63, 
1864, 
I  865, 

1 866, 
1867, 
1868, 
1 869, 
1870. 
.  1S71, 
1872, 

1873. 
1874, 

The  mean  correction  to  the  moon's  tabular  north-polar  distance  for  ciich  year,  from 
observati(ms  of  each  limb  at  each  observatory,  was  taken  with  a  view  of  detec-tiiig  any 
constant  error  of  sufficient  magnitude  to  alfect  the  final  results  for  errors  of  liie  iioth^ 
and  inclination.  These  means  should  have  been  taken  aller  the  application  of  the  cor- 
rections just  found:  actually,  however,  they  are  tlic  mean  corrections  given  by  the 
observations,  allcr  applying  the  following  constant  corrections  to  reduce  tiie  deilinalions 
to  tlic  same  fundamental  standard  : 


+  0.6 

—  0.1 

—  O.I 

0.0 

-0.6 

-f  0.1 

-0.8 

00 

—  0.1 

—  0.1 

-  '4 

—  0.2 

-  1.8 

-03 

2.2 

-0.4 

-  2.8 

—  0.6 

-0.6 

-3-8 

-0.5 

-4.2 

—  0.4 

To  Oreenwicli  obacrvationB  of  N.  P.  D. 


To  W.isliington  observivtiniis  of  X.  P.  I). 


1862-67,     —0.4 
1868-74,     -f  0.2 


1862-65,  -}■  O.^ 

1866-67,  —   I.I 

1868,  -  I   2 

1869,  —0.6 
(870-72,  —0.4 
•873-74-  0.0 

The.SC  corrections  are  approximately  those  necessary  to  reduce  flic  star-observa- 
tions of  the  several  years  to  Auwers's  standard  of  declination.  The  change  in  the  Green- 
wich correction  between  1867  and  1868  probably  arises  from  tin;  introduction  of  a  new 


33 


constiuit  of  refraction  in  1868,  \vliil(!  tlio  cliange  in  tlie  Washington  corrrclion  in   1866 
corresponds  to  the  introduction  of  tlio  largo  transit  circle  in  place  of  the  old  mural  circle. 


Year. 

Correction  to  N.  P.  D.  given  by — 

Greenwich. 

Washington. 

N.  L. 

S.  L. 

N.  '.. 

S.  L. 

1862 

ti 
—  0.1 

-  0.8 

II 

-  0.3 

-  0.8 

18&3 

+    0.2 

-  0.9 

-  0.5 

—   I.I 

1864 

+   0.4 

-  0.6 

+  0.8 

-  0.9 

1865 

+   0.5 

—  0.2 

+  1.2 

—  0.2 

1SC6 

-    0.7 

-  0.3 

+  1.4 

-  0.6 

1867 

-   0.4 

-  0.6 

+  0.1 

—   I.I 

1 868 

-    0.7 

—    I.O 

+  0.2 

+  0.2 

i86g 

—   O.l 

-  0.6 

-  o.S 

-   1.7 

1870 

-  0.6 

—    O.I 

—  0.1 

-   1.8 

1871 

—  0.2 

-  o.B 

+  2.1 

-   1.8 

1872 

0.0 

0.0 

-  0.7 

-  o.S 

"873 

-  0.9 

+  0.1 

-f-  2.0 

—  0.1 

1874 

•    • 

•    • 

-   1-7 

-0.5 

The  large  residnals  of  tl.c  Washington  observations  of  the  south  limb  led  to  the 
application  of  the  farther  syslematic  correction  ot  +  i"-0  to  all  those  observations  before 
conibining  them  all.  The  corrections  arising  from  the  error  of  mean  longitude  were 
then  ai)plie(l,  and  the  out.standing  residnals  were  considered  to  arise  from  accidental 
errors  and  from  errors  of  the  inclination  and  node.  The  equations  of  condition  thus 
betiomc 

0.92  sec  S  [sin  (/  —  6)  6i  —  cos  {I  —  0)  i  69]  =  dS 

or 

sin  (/  —  6)  Si  —  cos  (/  —  0)  i  SO  =  1.09  cos  SX^S 
Owing  to  the  smallness  of  the  final  residuals,  66,  the  factor  1.09  cos  6  may  be  consid- 
ered as  a  constant,  and,  in  the  actual  solution,  has  been  put  equal  to  unity.     lis  mean 
value  is  more  exactly  1.04,  and  its  ellect  may  be  obtained  by  dividing  the  final  results 
by  this  factor. 

The  final  values  of  the  residuals  were  then  arranged  according  to  the  values  of 
X—  9,or  the  moon's  mean  argument  of  latitude,  as  the  residuals  in  right  ascension  were 
arranged  according  to  the  mean  anomaly.  The  sum  of  the  residuals  corresponding  to 
each  interval  of  20'^  in  the  argument,  with  the  corresponding  number  of  observations 
for  each  year,  is  shov/n  in  the  following  table : 
5  M 


n^A-'Twmsmisfm^ 


I 


J 


II 


34 


Slims  of  errors  of  the  moon''s  eorrected  dedinafiou,  f/irrn  />//  ohserrntious  at  fireetnrieh 

and   Wiishiiif/toii. 


LimliSdf  ?. 

iSf, 

2. 

1S6: 
2.M 

N. 

186^ 

• 

186; 

i86(: 

. 

1867. 
X<J.I        N. 

1868. 

£i1.t 

N. 

1S3 

N. 

2.M 

N. 

£.!<) 

N. 

SiM 

N. 

o  to    20 

-   3.3 

8 

+  1.3 

3 

It 
+  4-0 

8 

+  5-4 

9 

+  26.7 

II 

-  2.5 

8 

+  0.4 

9 

20  to    40 

+  9'f' 

9 

+  5.8 

7 

-  0.4 

9 

+  Co 

7 

+  2.6 

12 

-  2.3 

9 

-  6.1 

•7 

40  to    60 

-   1-4 

9 

+  6.9 

10 

+  6.5 

6 

+  9-7 

7 

+  4-0 

9 

-  4.9 

10 

-  7.9 

«5 

60  10    80 

0.0 

7 

+  16.4 

10 

+  6.6 

8 

+  8.7 

12 

—   I.I 

16 

+  14.5 

II 

—  ".J 

5 

80  to  100 

4-   8.6 

II 

+  0.4 

12 

+  11. 1 

6 

+  7.9 

II 

+  5.5 

7 

+  0.5 

10 

-1J.8 

12 

100  to  1 20 

+  3-2 

7 

+  8.5 

15 

+  3.2 

5 

+  7-4 

7 

—   1.0 

8 

-  O.I 

6 

+  0.2 

6 

:2o  to  140 

-',.2 

12 

+  3.1 

8 

-  6.1 

S 

+  o.S 

II 

+  11. 9 

14 

-12.4 

8 

-  6.8 

II 

140  to  if)0 

-    0.3 

4 

-  4.6 

9 

—    2.2 

5 

-  9-7 

15 

-   1.2 

10 

-  7.7 

12 

+   6.2 

14 

160  to  180 

+    0.5 

9 

-10.4 

6 

-10.4 

12 

+  0.5 

9 

+  2.2 

10 

-  8.9 

9 

—  n.2 

9 

180  to  300 

-  8.6 

6 

-  5-7 

II 

-   0.6 

7 

-   5.3 

13 

-  7-7 

6 

—  15.2 

14 

+  3.1 

10 

2(XJ  to  220 

-22.3 

8 

—  II. 6 

10 

+  4.7 

12 

-  5.4 

9 

-  3-3 

ID 

-  6.8 

14 

-II. 8 

II 

220  to  240 

-14.4 

12 

—  10.2 

9 

-  8.8 

10 

—    I.O 

7 

—  2.0 

'3 

-  5.9 

12 

—  10.0 

13 

240  to  260 

-12.4 

7 

-12.3 

9 

-4.. 

8 

+  4.6 

II 

+  1.2 

9 

-  0.6 

9 

-  9-2 

15 

260  to  280 

-2.3 

4 

-   32 

4 

-  8.5 

8 

+  1.5 

9 

-  5-3 

9 

+    I.Q 

8 

+  1.9 

9 

2S0  to  300 

—    2.S 

7 

-  4.3 

8 

-  8.4 

II 

-  4.0 

4 

-  3.5 

■> 

-II. 4 

J3 

0.0 

«3 

300  to  320 

-  7.1 

10 

-  6.2 

10 

4-  9.6 

8 

+  3.1 

5 

—  0.1 

13 

-  8.4 

10 

+  0.4 

8 

320  to  340 

+  2.0 

7 

+  3-4 

8 

+  6.0 

12 

+  8.6 

6 

+  8.4 

II 

+  2.1 

<> 

-  6.7 

14 

340  to  360 

+  7.3 

5 

~  6.5 

5 

+  4.9 

13 

+  11. 6 

8 

+  7.0 

14 

+  2.9 

3 

-  3-1 

12 

—84.0 

142 

-75-0 

J54 

-49-5 

156 

—25.4 

159 

—  25.2 

191 

-93.1 

'75 

-85.9 

203 

+31.2 

+45.8 

+  56.6 

+  75.8 

+  69.5 
+44.3 

+  21.0 

+  12.2 

-52.8 

—29.2 

+  7.1 

+  50.4 

-72.1 

-73.7 

■iMM 


wimtm 


85 


Sums  of  vrrnrs  of  tin'  iimotCs  ainrctrd  (Icclhiutioii,  d'c. — C(»iitimic(l 

1873. 


1869. 


1870. 


1871. 


187a, 


Limits  ufX, 


1874. 


o  to    30 
20  (O    40 

40  Id  60 
60  tu  80 
80  to  100 
100  to  120 
120  to  140 
HO  to  160 
160  to  180 
I  So  to  200 
200  to  220 
320  to  340 
240  to  260 
260  to  280 
SSo  to  300 
300  to  320 
320  to  340 
340  to  360 


IMJ 

II 

+  7.1 

+  II. a 

+  6.4 

-  5.0 

!  -  2.0 

-  «3.7 

-  It. 4 

i  -  15.4 

-  2.5 

-  5.4 

-  5.4 

-  6.6 

-  18.4 

-  7.7 

-  11.4 

+  5.3 

+  5.7 

0.0 

-104.9 

+  35.7 
—  69.2 

7 

9 

II 

7 
9 
12 
II 
9 
4 
6 

7 

13 

7 

5 

10 


+  3-7 
+  6.6 
+  8.6 

+  3.5 
+  6.2 

—  6.2 
+  4.5 
-II. 7 

—  5.7 

—  0.5 

—  10.2 

—  I.I 

—  II. I 
-15-4 

—  10. 1 

—  9.' 
-10.3 

—  1.2 


><. 

ill,! 

7 

-  3.8 

10 

—  0. 1 

10 

-  0.8 

7 

+  13.2 

9 

+  13. 1 

12 

-  6.3 

7 

-  1.9 

II 

-  4.6 

13 

-  5.1 

6 

+  5.4 

12 

-  6.2 

9 

+  9.1 

S 

+  5-6 

15 

-  6.2 

5 

+  3.6 

7 

+  3-8 

12 

-  5.8 

6 

-  6-3 

4 
II 
II 

9 
9 

8 

7 

9 

II 

10 

•4 

8 

7 

8 
8 
9 
5 


Sil.S 

N. 

:;.i,i 

n 

It 

-  S.o 

C 

+  9.0 

-  7.0 

6 

-  7.7 

—  1.2 

10 

-  2.8 

-  5-1 

7 

-'3.7 

-  3.7 

8 

+  4.8 

—  2.0 

8 

—  I.O 

+  0.2 

14 

—  2.2 

-  8.9 

9 

+  7.4 

-  4.6 

12 

-  3.9 

-  Co 

3 

-  8.6 

-  2.9 

9 

+  5.2 

-  4.5 

10 

-  2-7 

+  13.7 

13 

+  14.8 

+   2.2 

II 

+20.7 

+  3-8 

9 

+  3.3 

-  1.3 

II 

+  4.3 

—  12.2 

12 

+  14.0 

+  0.3 

9 

+  7.2 

155 


-92.6 

+  33.* 

166 

-47.1 

+  53.8 

-59.5 

•1-  6.7 

153 


-67.4 


-47.2 


167  ;  —42.6 
+90.7 


+48.1 


It 

9 

+ 

7.7 

13 

10 

- 

7.5 

7 

- 

17." 

11 

' 

- 

25.4 

14 

- 

5.3 

6 

- 

22.4 

12 

4 

+ 

2.1 

6 

- 

12.6 

II 

- 

5-9 

10 

- 

7-2 

3 

- 

'9.3 

12 

- 

15.2 

6 

12 

- 

4.6 

6 

9 

- 

3.5 

6 

10 

- 

5-5 

13 

S 

+ 

0.3 

lo 

7 

+ 

4.0 

10 

II 

- 

3-9 

9 
169 

40 

- 

154.3 

+ 

14. 1 

140.2 

The  goiit'fiil  irregiihu-ity  of  the  residuals  in  (lecliiiiiliou  is  such  that  no  great  ad 
tage  woulil  result  in  a  separate  solution  ot"  the  equations  for  the  separate  yetirs. 
sum  of  the  residuals  for  each  20^  of  the  argument  was  therefore  taken  during  the  w 
thirteen  years  of  observation,  with  the  following  result: 


van- 
hole 


x-e 

iAr! 

N. 

A-O 

SAct 

N. 

0       0 

„ 

0 

,/ 

0  to  20 

+  47-2 

103 

I So  to  200 

-  62.3 

no 

20  to  40 

+  10.7 

115 

200  to  220 

-  95-3 

12S 

40  to  60 

+  6.1 

119 

220  to  240 

-  73.3 

126 

60  to  80 

+  12.3 

124 

240  to  260 

-  32. s 

127 

So  to  100 

+  34.3 

121 

260  to  280 

—  23.8 

106 

lOO  to  I20 

—  3f'.2 

III 

2S0  to  300 

-  50.1 

123 

120  to  140 

-  27.4 

120 

300  to  320 

-  5-4 

115 

140  to  160 

-  65.3 

124 

320  to  340 

+  19.2 

122 

160  to  180 

-  65.4 

126 

340  to  360 

+  20.2 

no 

.1 


^.m^f^fiKm^.-imtMiUi 


-~^—- ■—^r~——^,i-Li^:±jMiiM^At, 


|w| 


Ti<'uviiij(  in  tin;  ('(|iiiilinii«  a  ntiiHlni\l,  Irrni  '"i/i,  n'j»n'si'iitiiiff  (lif  inniii  roiisluiil  crinr 
still  oiilstatiiliii^  ill  tlic  iiicasiircs  of  tlt'diiialiDii,  tin;  solution  of  tin;  ('(|iiatioiis  of  coii- 
<litioii  given  \>y  tin;  residuals  gives  the  l()llowing  results: 

Jl>--n".i7 

Ji,  =  +  o".  I  5 

iJO-  —  iy"..\o 


or. 


Correetioii  to  the  iiielination,  — o".i5 

Curr(!elioii  to  the  longitude  of  node,  +4". 5 
This  correction  to  tlie  longitude  of  the  node  from  Hansen's  tabl(!S  iniplies  a  diiiii- 
luition  of  the  seeiilar  retrograde;  motion  of  the  node,  which  is  (|uitr;  aecor<lant  with  the 
results  derived  from  ancient  eclipses.  Hansen  remarks  that  an  increase  of  12"  per  cen- 
tury in  the  longitiuh;  of  tli<;  moon's  node  will  improve  the  agreeni(;nt  of  his  tables  with 
ancient  eclipses;*  and,  if  wesui»pose  the  tubular  longitude  of  the  node  to  have  l)e(;n  (;or- 
reet  in  1825,  this  would  imply  a  correction  of-f5".2  to  the  longitude;  of  tlu;  node 
in  1868. 


'  l)iirli';;iiiijj,  t'tL'.,Tli.  il,  p.  y)l. 


If       i 

^^  W( 

4     , 

37 


HH 


Al'Ml.lAKV  lAII.KS  R)K  !•  \(  I  I.I  I  A  11  N  ( ;  rill',  (( )M  I'l  I  \  IK  ).\  oh  llll,  CoKkir 
•I'lONSTO  IIANSKN'S  '-rAIILKS  Dl',  I, A  I.INK  ",  (ilVKN  \\\  IIIK  I'K  lai  I  >l  NC  DIS 
CUSSION. 

Tlio  loUowiiiif  is  a  siiintuiirv  of  Ww,  ('(trrccrKtiis  to  (lie  l(iii^itii(l(!  id'  llic  inooii  riom 
IIiiiiscm's  lul)lcs  1,'ivcti  by  llic  preceding  disciissioii.  I'iie  first  six  leniis  an;  iippliciililt! 
(()  tli(!  (lisliirhed  mean  Ningilnde,  or  '^'■Ari^itmen/  foiifliimr/i/ii/";  Ihe  remainder  lo  Hie 
(rue  loiigididi!;  but  tiic}'  may  all  lie  used  as  correclions  of  the  "Ji^mnr/il  foudaiiiinlnl" 
widioiit  serious  error: 
Concclioiis  on  accuitiil  of  (liiiiinution  of  the  sular  pdntl/nx .  .   n  <5c  z=  +  o" .(.)6  sin    I) 

-\-o".o7m\{l)-g) 

On  (icrount  of  /iif:r)//u'xis  [lure  in'orisioiHillij  sef  oxldr),  that 
the  moons  atilcr  of  i^rorily  t/oc^  not  coincide  with  the 
center  of  figure,  together  with  the  correction  to  the  erec- 
tion resulting  from  llie  correction  lo  the  eccenlr'iciti/ .  .  .  n  i^z  rz  +  o".09  sin  g' 

—  o".3,^  sin    :  I) 

—  o".2i  sill  (2  I)  —  ii) 
On  iiccoiiiil  of  term  nceidenlallu  iiitroiliieed  into 

the  tiihles  with  a  wrong  sign ('ir  —  —  o".62  sin  {2  g  —  4  g'  -{-  2  m  —  4  <>>') 

On  account  of  correction  to  the  eccentriciti/  ond  perigee 

found  from  observations  during  i  S47-74 6r  zz  —  o".57  sin  g  —  o".23  cos  g 

=z      o".62  sin  (g'-f  202'^.o) 
Empirical  term,  neceamri/  to  satisfy  ohserratio)/s, 

hut  not  verified  by  theory +  i"-5"  '*'"  [a'  +  2 1   .6  (  >'—  i  S65.  i )  J 

Unexplained  correction  to  the  mean  longitude,  changing  slowly  from  year  to 

year See  Table  IV. 

The  deduction  of  all  these  terms,  except  th(!  last,  has  been  fully  given  in  the  pre- 
ceding pages.  This  secular  correction  to  the  mean  longitude  has  been  derived  from  the 
outstanding  errors  of  mean  longitude  given  on  pag(!  30,  in  the  C(dumn  n  Sz,  l»y  suppos- 
ing this  quantity  to  vary  according  to  some  simple  law,  which  law  changes  Avhen  necessary, 
so  as  to  satisfy  the  observations  witliin  the  mean  limits  of  their  probable  error.  An 
examination  of  Table  IV  sliows,  that,  from  1848.0  to  1855.5,  t''*'  eornjctiou  is  supposed 
to  increase  uniformly  at  the  rate  of  o".20  per  annum.  It  is  then  supposed  to  remain 
constant  until  nearly  1S63.0,  a  period  during  which  the  observations  are  not  continuous, 
there  being  no  comparisons  with  theory  from  1859  to  1861  inclusive.  From  1863.0 
until  the  present  time,  the  observations  are  well  represented  by  the  corrcctioii 

—  5"-53  —  o"-S6(^— 1870.0) -fo".02  (^-1870.0)- 
The  continuance  of  this  correction  beyond  1875.0  is,  of  course,  purely  conjectural. 

TAr5I,ES  FOR  APPLYING  TIIK    PRIXKDIXG  CORRECTIONS. 
The  following  tables  are  designed  to  facilitate  the  computation  of  the  corrections 


mmm 


m 


.■■} 


M 


38 

just  <.'ivfi).  To  avoiil  tlic  iicccs.sity  of  lelcniii^  to  Huiiscir.s  (aides,  the  valiU's  olall  the 
necessary  argimieiils  arc  yiveii  lor  the  years  1850  to  18S9  in  'lalth's  I  to  111. 

Talile  I:  tlie  epochs  are  January  o,  Greenwich  mean  noon  of  common  years,  and 
January  i  »d'  leap  years.     All  the  ar^fumeids  increase  iiniforndy  i)y  a  unit  in  a  day. 

Ar^rnment  g  is  the  moon's  mean  anomaly,  converted  into  days  hy  dividing  its  ex- 
pression in  deirrees  l)y  13.065.     It  is  equal  to  llan.sen's  argument  g  diminished  l»y  15  days. 

Argument  D  shows  the  number  of  days  since  mean  new  moon,  or,  it  is  the  mean 
departure  of  the  moon  from  the  sun  expressed  in  days.  It  is  ecpial  to  Hansen's  argu- 
ment ^^  diminished  by  30  days,  or,  which  amounts  to  the  same  thing,  by  o''.47. 

Arirument  A  gives  the  number  of  davs  from  the  time  when  the  anjjle 

2g  —  4g'  +  2a)  —  ia)' 
was  la.st  zero. 

Arirnmeiit  li  is  that  of  the  empirical  term  indicated  by  observations,  but  not  given 
by  theory. 

Ar<rnment  u  is  that  of  latitude,  or  the  number  of  days  since  the  mean  moon  last 
passe«l  her  ascending  no>l('. 

Tiddes  II  and  III  do  not  seem  (o  U(.'ed  explanailoii.  In  using  the  former,  ean;  must 
l»e  taken  1<i  diminish  by  ou(!  day  tiie  dates  for  Jauu.iry  and  ^"ebruary  of  leap  years. 

Talde  IV  gives  tht;  secular  corrections  to  the  mean  longitude,  or  to  n6~,  obtained 
from  ol)servatioiis  in  the  manner  aln^ady  described. 

Table  V,  argument  J,  gives  the  correction  for  the  t(.'rm  introduced  into  the  tables 
lescribcd  on  paye  o.     I 


witli  a  wrong  sign, 

tilde,  and  is  therefore  designated  as  6v. 


properly 


ippli 


Talde  \l  irives  the  empirical  term,  whis'li,  so  far  as  is  known,  may  be  a)t|ilied  to  the 


true  loiiiritude. 


Talde  VII  "fives  the  sum  of  the  terms  of  mean  hjngitude 


+  o".96  sin  1) 


o 


■OJ 


Sin 


I) 


-o".i3sin(7>  +  ir') 
4-o".09  sin  g' 


Tl 


le  sun's  mean  anomaly,  g',  having  a  ])erio( 


dof 


I  year,  the  sum  of  these  terms  can 


lie  expressed  as  a  function  ol' .'>  and  llie  nioidli,  and  is  given  in  the  table  for  the  middle 
of  each  monili,  and  for  each  day  of  J). 

Table  VIII  gives  (h(!  sum  of  (he  terms  ol  true   longKnde  which  depend  wholly  or 
partly  on  the  moon's  mean  anomaly,  namely: 

+  o".62  sin  (g  -f  202".o) 
+  o".o7.sin(/;  — 5-) 
—  o".2i  sin  (2  D  —  g) 

Tin-  sun;  of  the  terms  of  n  S:  are  to  bo  reduced  to  eorreclions  ol   the  'ongitude  in 
orbit  by  multiplication  by  (he  fad  or 

1  -f-  2  f  cos  r  -f  -^  c'-  cos  2  g. 
This  factor,  less  nnitv,  is  iriven  in  T.ible  IX. 


!^ 


39 


'or 


convenience,  the  unit  of  the  faclor  is  uniittctl  from  the  lahiilar  luiiiil 
necessary  to  adt 


icrs,  so  llial 


it  IS  only  necessary  to  add  the  product  /  X  «  '^-  i"  ""i<h  n  S:  and  Sr  to  hav(>  tiic  cor- 
rection of  the  true  longi(nd<'  in  orlnt. 

These  corrections  being  applied  to  llie  longitude  of  the  moon's  center  from  Han- 
sen's taldes,  that  longitude  may  he  regarded  as  correct,  exce[)ting  a  small  correction, 
wiiicli  may  probably  be  regarded  as  constant  during  any  one  period  not  exceeding  six 
months,  and  which  may  be  attributed  to  tiie  adoi)ted  position  of  the  e(piinox.  It  will 
l)e  best  determined  from  occultations  of  stars  observed  at  points  whos(!  longitudes  from 
Greenwich  are  accurately  known  by  tehigraph,  and  will  then  be  applicaljle  to  the 
determination  of  the  longitude  of  any  station  from  occultations. 

If  the  corrections  here  deduced  are  applied  to  the  (,'rrors  of  the  lunar  eplunneris 
derived  from  meridian  oliservations,  it  must  bo  remend)ered  that  thesi;  observations  are 
made  on  the  moon's  limb,  while  the  corrections  are  applicable  to  the  center.  Hence, 
the  value  of  the  moon's  semi-diameter  must,  if  great  refinement  is  aimed  at,  be  varied 
with  the  ob.servef,  the  instrument,  and  tin;  brightness  of  the  sky.  For  large  instru- 
ments, Hansen's  semi-diameter  is  about  i"  too  great,  even  at  night. 

The  sum  of  all  the  terms  of  n  S:,  Sr,  and  FX"  ''^-  f'''>"^  the  tables  will  l)e  the 
correction  of  the  longitude  in  orl)it.  Tliis  will  not  be  rigorously  the  same  as  the  correc- 
tion to  the  ecliptic  longitude. 

Table  X  gives  the  small  factor  (F.  I)  liy  which  the  orl)it  longitude  must  Ijc  increased 
or  diminished  to  ol)tain  the  ecliptic  lonirituile.    This  tiictor  may  g(.'nerally  be  disregarded. 

Table  X  al.so  gi\es  the  data  tor  the-  correction  of  the  moon's  latitude,  namely,  (i) 
a  flictor  (i\ /?)  by  which  the  correction  of  the  moon's  argument  of  latitude  must  be 
multii»lied;   and  (2)  the  term 

•^/^  =  —  o".  I  5  sin  u 

arising  from  tlie  correction  to  the  t.djulav  ni,-lin.uion  of  <lic  moon's  orliit.  The  correc- 
tion of  the  moon's  arguaient  of  latitude  being  that  ol'  her  lon:;itu(le,  miinis  the  correction 
of  her  node,  the  wlude  correction  to  the  latitude  will  l)e 

dft-dft,  +  (F.ft)  {6l--^".s-) 
Table  XI  gives  the  factors  for  converliuL'  correction'-'  of  loniritude  and  latitude  info 
corrections  of  right  ascension  and  declinatio!:.     The  nn  ninhv  are 

S.Al     —  or  +  (f  .  a)  Si-  +  (/? .  a)  6/i 
S  .  Dec.  =:  S/i  +  (/• .  S)  Sr  f  (/?  .  S)  Sft 

The  side  argument  is  the  moon's  longitude,  auil  in  the  coefHcients  iv .  n)  and  perhaps 
(/y .«)  regard  must  be  had  to  the  moon's  latitude  also      Tlirci;  columns  are  therefore 


given  for  latitude, 


and  +  5     respectively. 


mmm^ 


40 


-'■  I 


As  an  example  of  the  use  of  the  tables  of  corrections,  we  will  commence  the 
deteniiinatiou  of  the  corrections  for  Scsptembcr,  1S74.  AVe  find  the  values  of  the  argu- 
ments for  September  i,  from  Tables  I  to  III,  a.s  tbllows: 


g 

D 

A 

li                u 

1 

1874  .    .    . 

Sept.  I     .     . 
I'eriDtls  .     . 

Arg.  Sept.  I  . 
A  TR.  Oct.  I  . 

5..» 

23.6 

-27.6 

12.  t 

7.8 

S.o 
1.9 

20.0 

24.6 

-27.4 

1.9 
26.2      i 
—  27.2      ' 

1.4 

19.9              9.9    1        17.2 

0.9      1 

3.3 

20.3 

1                                                                             ;                                                                              1 

39-9^!        47-2)'        30.0/: 
or     7.8  i^  or  19.8  )     or     3.7  )  ! 

1                    1 

rig.g 

:     1.4 

D-j;-. 

-18.5 

The  tabular  numbers  are  then  found  as  follows,  with  an  argument  incrc 
unify  each  day.     From  Table  VIII,  we  take  a  mt-an  from  columns  i8  and  19. 


a.-<ing  by 


September 


Ta 

lilc  1V(h,1 
VI  (,lr. 

VII  (.( 

fc). 

VIII  ( 

r)  . 

n< 

:XA,Tal,l 

■IX 

6v 

.      .      . 

ih' 

-4".5  ■ 

Ta 

lili:  X  (F  . 

/J) . 

(■'■ 

-4".5)(1^ 

/^). 

•Vi 

.      . 

I  —  9. II 
I  +  0.40 

—  1.07 

—  1.29 

—  o,f)5 


-  9. 1 1        -  9.11 

i 
+0.55        +0.62 

-  1 .29        —    1 .42 

-  1.25  -    I. 12 


0.72 


0.77 


-  9. II 
+   o  59 

-  1.50 

-  0,95 

-  0.7S 


—  9  12  —9.12 
+  0.4S  '  (-  0.28 
—1.48  1   —   1.40 

—  0.76  I   —  0.50 


0.75 


o.f)9 


—  9. 12 
+  ti.05 

—  1.23 

—  0-37 

—  0.60 


-11.72 


-II. 82 
-   0-93 


- 1 1 . 80 


■11.75 


—  11.6J     ,   —11.49 


0.71     ,   -  0.47 


-12. 84 


-17.3 
-+-  0.088 

-  0.03 

-  1.52 


J's  longitude 

(i +(».'>))  i!f , 
(,3.«).V  .  . 
AM    .     .     . 


(v .  il)  M'     . 
(I  ■<-/)..)) iVi 
iIDcc.  ,     , 


1-55 


46.5 
-i3.o8 


-12-75 


^17.2 

-t-  0.082 

—  0.07 

-  1.3S 


—  12.51  —12.22 


-17.0 


-16.7 


+    0.070       +    0.056 


O.  10  :      —      O.   12 


1. 19 


0.93 


0.28 


0.06 


—  11.27 
-1-  o.i3 


—  11.91 


-  IC.4 

-I-  0.038 

-  0.14 

-  0.63 


-11-55 


—  11.09 


-   I--I5 


Co.  7 


13.49 


-+-  0.47     I  +  0.32 


—   1.29 


74.6 


83.1 


-13.84     j  -13.81 

-1-0.16  !  -f  0.02 


0.77 


101.5 


—16.0      —15.6 
-f  0.019   ~  0.002 


-  0.15 

—  0.30 


0.15 

o ,  03 


9.12 
0.18 

I  .0! 
0.22 
0.48 


+  "-37 


-10.64 


-15-I 


—  0.14 
+  0.33 


-  0.45 


114-5 


-13- 


-12.6 

-0".84 

-  3-70 

-  1.50 


-  5-2 


-13-2 

-o'.88 


—  2.64 

-  1.42 


4.1 


-13-7 
—  o".  91 


-  1. 41 

-  1.23 


2.7 


•13-8 

-0-.92 


-  o,l3 


n-5 


.90 


— I2.6r     1  — 

—  0.09 

-12.7 

—  o'.85 


-t-   1.02        -(-2.05 
-  0.77       -  0.44 


+  0.2 


-j-   1.6 


+  3-5 


^ 


•tmmS*:s.sS3KSSS 


■^ 


41 

This  (..miMifalion   Ims   Ik.-.;,,   cm.fiu,.,,-.!   tu  .875,  Jai,u,-,-y  3 .,  an.!    ll.r    n..s,.lls  ;nr 
slKtwii  III  ilic  lollowiiiy  tal)Ie: 

Corrections  to  tlw  Ephrnirris  drrircl  from  Jfansnfs  Tables  of  the  ^foo^,   fhr  firrninhh 
inmn  ,»,.>„  of  cuh  itnij,  from  1S74,  Srpfemhrr  1,  to  \^--^,  .htnnor],  31. 


D.itc. 


Corrociion  to  latnil.ir — 


Date. 


Coircciicui  lo  i.ij.iihir- 


GM 


Gr 

.  nicm 

~ 

— 

I 

-T 

^- 



n 

oon. 

^:-i. 

Lonif. 

Lat. 

i    R.A 

Dec. 

tir.  nu-;  ii  i 
no„n.      I    ■-on,^^ 

Lat.        R.  .A. 

Dec. 

■  . 

^^ 

lS7(.       ' 

St 

r'- 

—  I2.S 

—   I. 

<■     —12, 

<>     -  ■;.: 

(JlI.      II        —    7.5 

+     '.I        -    '>.1        -r     3.6 

2       12^ 

I . 

'-          '3- 

2          4.1 

>2                7.2 

1.0            6.S   ^         3.2 

3 

12  5 

'• 

3           13. 

7           2.7 

13                ('.') 

I.O           6.S   '         2.7 

1 

A 

12.2 

I. 

'          >3. 

^      —   1.2 

14   i          6.6 

"•8  :         7.0           2.., 

! 

5 

II. 9 

0. 

5  ;       13. 

5      +  0.2 

'5  j         f).4 

^•7            7.0            1.3 

1 

t 

-W.I, 

—  0. 

5  j   —12. 

7      4-   1.6 

If)  i   —  6.2 

+   0-5      -   7.0      +  0.6 

7 

II. I 

—  0. 

I           II. 6            2   7 

17 

fi.i 

0.3            6.9  '   -   0.2 

3 

10.6 

+  0. 

:  i      10. < 

>  ,        3-5 

18 

6.2 

!   +  0.1            6.S            I.O 

9 

10. 1 

0. 

>  i         9.< 

>\        4.1 

'9 

6.4 

f  —  "•'            6.6            i.s 

r 

i 

10 

.J.6 

o.- 

j         %.i 

'  i        4-3 

1 

20 

1       „ 

6.8 

;         0.4           6.6            2.5 

II 

-  9.0 

+    C.f 

)      -  -■' 

)  '  +  4.4 

-  7-5 

—  0  6 

—  f'-O  1   —  3-3 

12 

5-3 

I.' 

7  • ' 

4.2 

22  j        3.3 

1         0.9 

7-3  ',         4-1 

'3 

7.'j 

1. 1 

''■: 

3.S 

23           9.3 

I .  t 

■^•i            4.7 

M 

6.3 

I.I 

6.2 

3-3 

24  !     I". 4 

'■3 

9-2             5.2 

■5 

6.2 

I.U 

5.Q 

1          2.8 

11         ''       ■'•■' 

I  .; 

10.6            5.2 

\U 

—    £.6 

+  0.9 

.   -   5-7 

+  2.2 

ii 

2(,        —  I  2  .  .4 

-     I.=    ■    -12.2        -    4..S 

>7 

5-2 

c.  7 

5   5 

1.6 

27            13.2 

'.4           13-9            3.8 

H 

50 

0.6 

5.5 

I.O 

ll               28          13.6 

1.2            15. I               2.3 

19 

5.» 

0.4 

c.? 

•i-  0.3 

29        13. s 

10            15-7—06 

20 

■•4 

+    0.2 

6.1 

-  0.4 

3"         13. f' 

0.6            13.3       4-    I.I 

21 

-  6.1 

0.0 

-   6.6 

—    1.2 

■'               31   1   -13-2 

-    ".3       -14.2        -r    2.6 

22 

7.0 

—    0.2 

7-2 

2.2 

■N''n-.      I           12.4 

+    0.1             12.6              3.6 

23 

S.I 

0.  5 

7-5 

3.2 

2          11.4 

0.4            II. I               43 

24 

9-4 

0.5 

8.5 

4.3 

ji                     1 

1                 3          I'l-S 

"•7            9-6            4.5 

2; 

»o.6 

I.I 

9-3 

5.2 

4   ;         9-5 

o.S 

8.4            4.5 

20 

—  II.S 

-    '3 

-10.3 

-    5.8 

5^-8.5 

+     I.O 

-   7.4      +  4.3 

27 

12.7 

'•5 

11.5 

6.0 

6            7.7 

I.O 

6.7            3-0  ■ 

25 

'3-4 

1.6 

12. S 

5.7 

7         7.1  : 

I.O 

6.3 

3-5 

29 

'3-7 

'•5 

«3.9 

4.8 

8            6.6 

1 .0 

6.2 

3.1   \ 

30 

13.? 

1.4 

M.9 

3.4 

9            6.4 

I.O 

6.2 

2.6 

Oct. 

I 

-13   5 

—   1.2 

-15.2 

-   '.7 

10  '  -    6.3 

+  0.9 

-  6.5 

+  2.1 

2 

130 

0.9 

'4.7 

-o.t 

II           6.3 

0.7 

6.9 

'•5 

3 

12.2 

0.5 

■       13.6 

4.  ,.4 

12          6.5 

0.6      7.3 

+  0.8 

4 

I  I  .  5 

—  0,2 

12.2 

2.6! 

13           6.8 

0-4  ;     7.7 

—  0.: 

S 

ro.6 

-t-  0.1 

10.? 

3-3! 

14           7.r 

+    O.I    1          7.8 

I.O 

6 

-   9-9 

-^  0.4 

-  9.4 

+  3.g 

»5  ,  -   7.4 

—  0.1   ,   _   7,8 

-   1.3 

7 

9-2 

0.6 

8.4 

4-1 

16 

7.7 

■-'•4            7.7 

2.7 

8 

8.7 

0.3 

7-7 

4.2 

'7 

8.1 

( 

0.6           7.6 

3.4! 

9 

8.2 

I.O 

T.2 

4.2 

18 

8.4 

0.8 

7.6 

4.0 

to 

7-8 

I.O 

6.8 

4,0 

19 

8.9 

I.O 

1 

7.8 

4.5  , 

•t ' 


i 


St 


42 


M  ' 


»4 


'#? 


,.„,.,.,.,„■„„.  ,„  <,..■  r:,.i. :^  .i^rirM  /,■ //»»«■«•»  ''«'■'■-  •:'  "-  ^' "-<■---'• 


[);ilc. 

Cil.   HUM 

■       i?74- 

Nciv. 


I 
I.uMK.        Lai.  K.  A-        n^'<--- 


20  i 

21  ; 

I 

22  I 

'-3 
24 

25 


Dec. 


2S, 
2.) 

3" 
I 

2 
3 
■1 


5 

6  ^ 

7  ; 

8  i 


10 
II 
12 
13 
14 

15 
16 

17 

IS 

19 

20 

21    i 
22, 

23  i 

24  I 

25 
2f) 


-    <)-4 

1 0.0 

10, f) 

1  I  .2 

I  I  .7 

-12.2 

..2.5    I 
12.f) 

12.5 ; 

[2.  I 

-11-7 

II. II 
10.3  i 

94 

S.5 
-  7-7 


6.4 
6.1  'j 
fi.o 

-6.1 
fi.4 
6.S 
7-4 

S.o 

-  8.7 
9'3 
')■')  , 
10.3  ', 
10.7 

—  I  I .  o 
11.2 
I1..1 
11.4 

II. 4 

-II. 3 
i       II. I 


-  I . 

1.3 
1 . 3 
1-3 
1 .2 

-  1.11 

o.fi  ■■ 

I 

-  0.3  . 

(l.O 

h  ".4 

+  0.7 

I  .1  I 
1.2' 
1  .2 

+    I.I 

I  .0 
U.() 

'        0.8 
'        .).fi 

+  0.4 

+  0.2 

0.0 

-  0.3  : 
o.fi 

-  0.8  I 
1 .1  i 
I  .2 

1-3 

1.4 

—   1-3 

1 .0 
0.7 
0.4 

—   O.I 
+  0.3 


-    S.2 

m.  1 


11-4 

12.7 

—  13.S 
14. I 

'3-^ 
13.(1 
ii.y 

-ID. J 
')■') 
')■" 

.S.2 

7-5 

-   7-" 
6.8 
(i.f) 
6.5  I 
6.7  I 

7-3 
7-5 
7.f) 


-   4.S 
4.3 

3S 

2.1' 
—     1.2 

+  "'S  ;, 

2.0  \\ 

I; 
3-3  1; 

4.3!! 
-t-  4.<)  !| 

1 

5 -Ml 

11 

J 
-f-  3.6  ii 

3.0 
2.3 

o.g  « 

+  0.1    , 

I  ~  °-"  i^ 
1.5   :. 

2.4 

3-3 


DaU'. 

Gr.  mean  ! 

nnoii.     I 

i 

1374-      ' 
Dec.     27 


Coiroclioii  Ici  laliiilai  — 
Long.         L^t.        R-^-        ^'=''- 


10.8 
10.4 

•').  I 
.  6 


I.m. 


-  7-8 
S.l 
8.6 
■).2 
10. 0 

—  II.O 

12.0 

12.7 
12.9 
!      12.6 

—  12.0 
II. I 


I 


4-7 
5.0 

5-' 
4.8 

-  4.1 

31 
1-7 

—  0.2 

+    1-3 
+  2.6 
3-<>  I 


3  ; 
5  ! 

7 
III 


-  8.7 
8.2 
7-8 
7.4 
7-1 
-  6.,) 
1  6.8 
6.., 

7.6 


4-  0.6 
o.S 
l.o 
1 .2 
1 .2 

+  1.2 
I  I 
1 .0 


—  10.2 

')-5 

,5.() 
S.4 
S.  1 

-  70 
7') 
7- 9 


1 


'3 
14 

I  5 

16 
17 
iS 

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48 


TAiii.i.;  VIM,  <Sr. 

/Jcrizoi./ti/  Ai;^iimriil,  or  . hxioiiiii/  ill  A'/.  /■>— ,c,  <'/■  /v*— ^•  +  30.      / \-rt'u\il  Aixiiiiinit,  1;. 


i 


A 


I   ;. 


1," 


I 


O 
I 

2 

3 
4 

5 
6 

7 
8 

9 
10 
II 
12 
13 
14 
15 
16 

17 
IS 

19 

20 
21 
22 
23 
24 
23 
26 

27 
2S 
29 
30 


■O, 

— o. 

-o 

-o 

-o.. 

-o. 
— n. 

—  o. 
•0. 
■0. 

—  o. 
— o. 

-o. 

o. 
-o. 
-o. 

o. 

o. 

o. 

o. 
+0. 

i  "• 
o. 
o. 
o. 

-)-o. 

h<.  . 

o. 

— o. 

-o. 

—  o. 


23 

I 
39 

54 
fjf. 

75 

r^ 

So 

78 


— o 

—  o. 
— o. 

—  o. 
— o. 
— o. 
— o. 
— o. 


"' 

— 0. 

62 

— 0. 

49 

-0. 

33 

— 0. 

IS 

— 0. 

00 

+0 

>7 

0 

32 

+0 

46 

0 

5S 

0 

67 

° 

73 

0 

74 

+0 

70 

0 

64 

0 

55 

0 

45 

0 

2.) 

+0 

15 

+0 

02 

— 0 

17 

— 0 

3' 

— 0 

45 

— 0 

30 

60 
71 
77 
81 
Si 
7ft 
6S 

.50 !— o 

( 
41  j-o. 

26  i— o. 

I 
oS   +0. 


—  o. 
— o, 
-o. 

—  o. 
— o. 

—  o. 
— o. 
— o. 
— o. 


10 

2f) 

42 

56 

63 

76  i  o 
80  j  o 
7(}'+o 


o 
o 
+0 
o 
o 


i 


.67 

•  57 

•  43 

.25  J+-0 
.11  '  f  o 

J 

.06  —  o 
.22|— O 

.38J-O 
•52I-O 


36  — o.3()  j-0.31) 
5"  -0.53  —0.50 

6j    -0.621—0.58 

I 
7-.;  — 0169;— o.6| 

77|-"-74!-o.fi7 
So'— 0.74  i— 0.66 

I 
76  —0.70  —0.60 

i  I 

71  I  — 0.62  1—0.52 

61  i— 0.51  1—0.    2 

47   —0.33   — 0.2S 

33   -o.22:-o.i3 
lO  ,—0.06  1+0.03 

03    +0.  II        O.  Iq 

i 
.  I9j     0.23I     0.35 

.36'     0.45  I     0.50 

.52;+o.59i  (-0.63 

I  1 

.65      0.71^     0.73 

.75  I     o.So 

.82;     0.84 

.S3:     "•84 

.  =1    +-0.S0 

■  1''  +"-73 
.67  f-o.62 
•  54'+-o.49 

■  3)   +0.33 


0.79 
0.S2 
o.Si 
(0.76 
o .  67 

0.56 
0.42 
0.26 


22    -l-o.  16  j +0.  10 
.06    -o.oi  i— 0.09 


12 

—  0.20 

-0.26 

30 

-0.37 

-0.  |2 

46 

-0.52 

-0.^7 

60 

-0.66 

—  0.69 

1 

-0.35, 

I 

-0.44 

-o.52| 
-0.56J 

— 0.5S 

-0-55 
-0.50 

-0.43 
-0.32 

—  0.20 
—0.06 
-1-0  09 

0.24 
0.3S 
0.51 
+0.62 
0.71 
0.76 

0.77 

0.75 

f-0.69 

0 .  6n 

0.4a 

0.3; 

n.2') 

•)-0.02 
i 
-0.14 

—  o .  30 

—  o  46 

-0.59 

-o.6f, 


-o.aS 
—0,36 

-0.43 
-0.47 
-0.48 
— o  46 
—0.42 
-0.31 

-0.26 
-0.14 
—0.02 
(-0.  II 

0,25 

0.37 

"•49 

+  0.59 
0.66 
0.6S 
0.69 
0.67 

+-0.61 

"■53| 

I 

0.421 

+  0.  13 
—0.01 
-0.16 
-0.31 
—0.46 

-"•54 
—0.60 


— o.  19 
—0.28 
-".34 

—  ".37 
-0.39 
-0.311 

-"■35 
-0.30 

—  0.22 

—  O.  12 

—  0.02 
t-O.  10 

0.21 

"•33 
"•44 

HO.  53 
0,5  s 

o .  60 
0,62 

"■59 
f  0.54 

0.47 
0.37 
0,21 

+  u.  -2 

—  0.02 

—  o    16 

-,).3n 

-"39 
-0.47 
— "■  54 


-o.  1 1 
— 0.I9 
—0.25 
—0.31 
-"•34 
-"•31 
-"•33 
— 0.2S 
—0.22 

-0.15 
—0.06 
+  "."l 

O.  1; 

0.2l'i 

"■37 

|  +  "-43 

!  0.48 

"•53 
"■54 
0.53' 

,+-0.49, 
0.43! 

"■33 

n.2| 

0.13 

■fo.oi 

- -"■13 

—  0.22 
-0.31 

-"  3); 

-"■44i 


'> 

10 

II 

12 

•3 

«4 

-0.03 

4-0.02 

+0.03 

o.no 

—  (1 .  nfi 

-n.  1; 

-0.12 

—  0.09 

—0,09 

-  0 . 1 .1 

—  0.22 

-n.31 

—0.20 

-0..9 

-0.21 

-0.27 

-0.35 

-0  45 

-0.27 

— n.26 

-0.31 

-0.38 

-0.47 

-",58 

-0.31 

-"■33 

-".39 

-0.46 

-0.57 

—0.67 

-"•34 

—0.36 

-"•43 

-"•53 

-:>.64 

-"•74 

-"•33 

-0.3S 

-"•47 

-"■57 

-0.67 

-0.76 

-0.31 

-0.3S 

-"•47 

-"•57 

-0.67 

-"■73 

-0.27 

-0.35 

-0.45 

-o.;.| 

—  0.61 

-0.6.^ 

—0.21 

—0.30 

—  0.3S 

-o.)7 

-"■53 

-0.53 

-0.13 

-0.22 

—0.30 

-"■37 

-".44 

-0.4S 

—0.03 

—0. 12 

— 0. 19 

:-o.23 

-0.33 

-"•33 

(-0.07 

—0.01 

— 0. 10 

.-0.17 

-0.18 

— 0.  IS 

0. 19 

l-o."9 

0.(«) 

'-0.03 

-0.04 

—0.01 

0.27 

0.17 

4-0.13 

'+0. 10 

-fO.II 

4-0. 14 

*-"34 

f0.2S 

+0,23 

+0.23 

-:-o.24 

-(-0.29 

0.42 

"•35 

"•33 

"•33 

"■37 

"•43 

0.46 

0.42 

0.41 

"•43 

0.4S 

0.56 

0.49 

0.46 

"■47 

0.50 

0.57 

0.65 

'     0.4S 

0.4S 

0.49 

0.56 

"•63 

0.72 

,+0.47 

+0.46 

4-0.51 

+  0.5S 

-(-".67 

+  "•73 

'     0.41 

0.41 

0.51 

0.58 

0.65 

0.71 

"•35 

0.40 

"•47 

<--?3 

0.60 

0.67 

H.27 

,     "^33 

0.33 

0.4S 

"•55 

0 .  60 

0.17 

0.22 

0.31 

0.40 

0.46 

0.50 

+0.05 

■4-O.I4 

+  0.24 

+-0.31 

-l-o.jft 

+  "•37 

—  0.1)4 

-j-o.ofi 

0.13 

0 .  20 

0.23 

"■23 

-0.13 

-0.<)| 

f  0.04 

+o.of 

+  0. 10 

-fo.oS 

—0.22 

-0.13 

— cr.07 

—0.02 

-o.oi 

• 

—0.07 

—0.29 

—0.22 

-0.15 

-0.13 

-0.15 

—0.20 

-0.36 

—0.27 

1—0.23 

—0.23 

—0.26 

—0.32 

, 


^"^■Mf  »■'!'?" 


I  ■ 


41) 

Taum;  \'I  1 1,  '"i^ — ( '(Hitiiiiicil. 

lloli.oillill .  h.;iiiiirilt,  or  .  Ir^iiuii'ii/  ,it  l,f.  /'  — -,  or  /;_ -4.  ^o.       /  i  rlioil  .  hxiiiiirii/,  -. 


15 


I'l  17  |S  I, 


■Ji  -M 


■J.J  y> 


0 

-0.25 

-0.35 

-0.41 

-d.  )S    -0.4S 

-  (1.4(1  — o.4r 

'—0.31    --(1.23 

d  ,  1  1     - 1 1   d, 

-0.117  -0   oS  — d.  1  J       .11- 

11  Jii 

I 

—0.41  -i),5i  —0.58  -II. fiii;  — o,(i(> -o.5f.— 0,48 —0.40  -i(  1(1 

—  0.25  -  O.SO— 0.2(l  -  0.22  -0   2.S  -d.3: 

-d  42 

2 

—0  56—0.64  — o.fig— 0  7tl~r).fifi:  — 0712—0.55—0.45; —0.  3() 

-"33-"  31 

!— 0.3s    _0.3(l;-0,    (3    _0     51, 

-0  57 

3 

--0.6S 

-0.  7t  — 0.  71)  — (1  7S  -0.74    -<i.(i7  — 0.  5S  — 0, 51    -0.44 

-d.  |Oi-o.3()|— 0.42  -0.  17    -d  ;.-      d  1  i 

d.7d 

4 

-u.7( 

-o.Sj  -0.83  -<).,S  1-0.71   -0,67  — o.f)0  —0.52,  -0.47 

-0.45  ~o.  |(:-o.  511    -d.;7       0.(15    -"■7' 

-"■77 

5 

-0.81 

—0.84-0.84   -0  8(1— 0.74  -o,fif)!-o.5S  — o,52|-(i  4S 

-0  4S  -d.  ;,(■  -d.  5(1      11  (u:— 0.72  — 0.  7.S 

(i.So 

6 

-o.Si) 

— 0..S3  — o.Si  — 0.  761— o.<i(j  — o.fii  -(1,5  J  — 0.411'— 1)  47 

-d.  17  -0  52  — o.i.ii  -(i.(iH  _i)  71    -d.  7- 

11. 8| 

7 

-"•77 

—0.77  —(1.74  --0.68,— 0. to  -0.53  -0  47  -0.43—0.42 

-o.4'i  -11.52  —d. (id —11.(17    -d.73      (1.77 

d.  77 

8 

— o.6() 

—o.h)  — o.d|  — 0.571  _o.  501—0.43  — 0.31)  — "•3f:-o.37 

-d.42   -0.  5d  —  d.  57  —0.(13  —  d.(ii|  —0.71 

-i>.7d 

9 

—0  60 

—0.56  —  (\  51  —0,43  -0.37    -(1.31  — 0.26  -0.  2(  —11.  30 

-().3(.  -d.43  — (1.51  -0.58—0.(12   -d  (12 

-<>.5() 

10 

-o.4^ 

—0.42  —0.35  —0.2'^  -(1.22  —  (1. 16  -d.  [4  -d.  ifi  -0.21 

-d.27  -0.35  —0.  13  -d,  |i,  -0.52  —0.  51 

-11.  15 

II 

—0.30 

—0.2?  —0. 1 ()—'».  1 2   -0,05  —  (i.di  —0.02  —  0.04  —(1.  Id 

-d.  iS,  -0.27  -0.  34  —11.311:  -0.401—0.  3(1 

-11.21) 

13 

-0.13 

—0.08—0.01  +0.07  fo.12  K).  r3  +  >.i2  fo.o"*     o.dii 

—  o.di)    -0.17  -0.23  -d.  2(1    -11.2:  -d.  2d 

11   13 

13 

+0.03 

4-0. 10  f  d.  I)- 

d.2J!      0.27       (1.28       d.2C       d.lS    t-o.ld 

f  0   01    -0   0(1—0.  12  —0.  13— 0.  10    -o.dj 

1  d  115 

14 

0,20 

0.2S      0.35 

0.411      0.43      (1  41      d.3f      0.2"      0.  Ill 

0.  10! 4-0. 03!      o.diM  d.dl|  -t-O.O^'H-O.  121 

d.  22 

15 

+  0.37 

+  0  44  +0.51 

4-0.55, +0.5;   to.  51  -f- 0,44  -f-0.35  fd.2(i 

4-d.  1  -  -Ki.  131 +(1.  I  1    I  1),  1  ■;  -I  (1.  In   K).2-^ 

!  11.  3(1 

16 

0.51 

0.5S 

0.6= 

o.fjfi     o.fii      0.5S     0,50     <j.4i!     0.33 

0.2-      0.21J     d.2l      0.25:     0.32      (1.411 

d.  51 

'7 

0.63     0.71 

0.74 

(I.73      d.fll)     0.(13     0.54J     0.45      0.3C1 

d.3d      0.28      d.2ij      0.35      0.42      11. =2 

■■.(■3 

iS 

0.73     0.7S 

0.7S 

0.77      d.72      d.64      '1.5  = 

11.45       0^38 

".33      d.331     o^'i     ".41!     o.;i      d,(i2 

■•  71 

I') 

0.77      o.Sti 

O.Si 

0.7'"      d.71      d.'i2      d.52 

0.43,     "37 

"•31      "3= 

0.311      0.47:     0.57      o.d,'* 

(1.7(1 

20 

+  "■77 

+  o.Sd 

+  o,7() 

4-0.  7  1  -f  0  fill 

+-0.5(1  -t  0.4' 

-i-o.3(>'-l-o.3(  +0.32  -t-o. 3' 

•4-0.40  i-d.  511  +(1.(111   (  d.  71 

!  d.77 

21 

0.76 

o.^h 

"■73 

"■(>r    0.57 

(J. 47     0,31'     0.31      0.2S 

0.28      0.32 

d.  jd      d.  51'      d.  ;i)      d.  (i,^ 

".-3 

22 

a .  70 

0.68 

().fi4 

0.55;    0.4; 

0.3(1       d.2.*' 

0.23.      d.2d 

d.22,     o.2^ 

0.37      d.4('      0.55      0.62' 

11.(1(1 

23 

o.fio 

0.5S 

0.51 

0.42      0.32'     0.23      O.lfi 

-ho.ll      o.ii 

11.11;     0.22 

0.311      0.411      0.4,^      0.5-! 

"•57 

24 

0.40      0.43 

"3= 

0.2(1  t-d.  i(  ,-f  ii.dS  +o,or 

—  0.01    -Ki.dl 

T-d.d'i        d.I" 

0.22      d.31       d.3>      0..13 

"•  14 

25 

1-0.34  fo.27 

rd.K) 

hd   0  1        d.di  '  — d    d7    — d.  1 1 

-0    14  — d.  1  1 

—  d.115    +-".d|    -|-d.l2    t-d. 2(1   -l-d.27    )il.3( 

t-d.  21) 

2f) 

0.  iS  +0. 1 1 

l-d.dl 

—0.08-0.171-0.24  —d. 271-1). 2(1  -d.23 

—  0.  15  —  0.(17  -i-d.d2   4-1).  Id    f  (1.  14    4-11.15 

r  d.  12 

27 

•t  0.02  — O.llfl 

-d.ll-. 

0,25  —0.31;— d.31)  ~d, 41 

-0.311-0.33 

—  d.  26  —  d.  1(1  —  (.1.  d,^  -~0.d2        O.du  —  d.dl 

-  (1.  di> 

28  , 

—  0.14  -0.23  —0.32 

—(1.42  —  d.411,  -d.  53  -  (1.  54 

— 0.50'— d.  43 

—  (1.3;    -d.2f   -        .  "1       d.  1  4  —n.  I  4    -d.  17 

-  11.  2d 

21, 

— 0.2S  —0.37 

-d.4S 

— 0.  57;— d.f)3  —  d.fif  -  0.(14  —0.  58 —d.  511 

-d.41    -1)    31   -.1    ■_  •        11.2(1    -11.27    -d.2i. 

-d.3; 

30 

—0.41,-0.51 

1 

—0.62 

.     _; 

-o.69J-o.73;-d.7i  -d. 70,-0. 64|-o. 55 
1             1                          III 

-0.471-0.41    -0.37  -d.37  -0.37  -0.42: 

1                                                      lit 

-"■4'J 

7m 


I-      « 


50 


I  Taiii.k  I\. 

'  Aii;:i'il<iil.  A'-   h'ltcli'i  1(1  hi 
iiiii//if'/i(il  I'V  It  "C. 


^■-. 


.  lixnniiiit,  II.  fihiiii  s/iir  ionrclinn  of  liilitliilc  ,iihl  itdiic 
lion  to  fcliptic  loii^iliiii(\ 


C) 

I 

+    I).  1  iS 

D.I  1   ) 

0 

3 

i>.li>3 

2 

3 

(i.i)Sfi 

3 

1 

(i.(/i5 

4 

5 

•t    i).o4c> 

5 

(. 

■k    11.1115 

(1 

7 

--     (i.(itH) 

7 

S 

-  0.034 

•S 

') 

—  o.(i;4 

'J 

lo 

—  0.(172 

10 

1  1 

—  o.oSO 

1 1 

12 

—     O.DIjf) 

12 

13 

—  0. lol 

■3 

U 

—  0. 103 

14 

"5 

-  o.oo'J 

15 

lO 

—    0.01)3 

ifj 

17 

—  o.uSo 

'7 

IS 

—  o.()'-3 

I3 

"> 

—  0.(140 

"J 

21) 

—  0.(124 

20 

21 

Y    o.ooi 

21 

22 

0.026 

22 

23 

0.05  I 

23 

24 

0.075 

24 

25 

+   o.(i(j4 

25 

26 

0.  lOIJ 

2fl 

27 

0. 1 16 

27 

28 

0.II7 

2S 

2.J 

0.  1 1" 

2J 

30 

+   0.01/1 

30 

(/■■./) 


— 

0 .  (Xl.( 

- 

0.(K)4 

- 

0.("'j 

-■ 

O.OOI 

•t- 

O.OOI 

+ 

o.(X)3 

0.004 

0,004 

o.0(J4 

+ 

O.0O2 

O.IXXJ 

- 

O.0(  • 

- 

() .  (K)3 

- 

o.(j04 

- 

0.004 

— 

0.003 

- 

o.(yj2 

0.000 

+ 

0 .  ( W2 

0.003 

4- 

0.004 

0.004 

o.(X)3 

-t- 

O.OOI 

O.CXX) 

— 

0 .  (X)2 

- 

0 .  (X14 

- 

0.004 

- 

0 .  (»4 

- 

0 .  003 

— 

0,001 

(/•••/'<) 


+     O.OI}() 

0.088 

o.(i.Si 

O.ofll) 

0.054 
■V  o  .136 
+  <  .017 

—  o.O(J4 

—  0.024 

—  0.044 

—  0.0(10 

—  0.074 

—  0.084 

—  o.oS(j 

—  0.089 

—  0.085 

—  0.076 

—  o.(jC4 

—  0.047 

—  0.028 

—  0.008 
4-  0.012 

0.032 
0.050 
o .  066 

+  0.078 
0.0S6 
o.oip 
o .  o8(j 
0.082 

+  0.072 


<'/5i 




0. 
(1. 

(Kl 
03 

- 

0. 

"7 

- 

0. 

10 

- 

0. 

12 

— 

0. 

14 

- 

0. 

»5 

- 

0. 

15 

- 

0. 

14 

- 

0. 

13 

— 

0 

II 

- 

0 

oS 

- 

0 

05 

- 

0 

02 

+ 

0 

01 

+ 

0 

05 

0 

08 

0 

11 

(J 

13 

0 

14 

+ 

0 

«5 

0 

15 

0 

•14 

0 

12 

0 

10 

+ 

0 

07 

0 

.04 

f- 

0 

.01 

- 

0 

■  03 

— 

0 

.06 

— 

0 

.09 

I  ii 


51 


Taiim-.  XI. 


luulois  J\>i-  (oiivciiiit:^  unall  iluin^.s  ,if  lon-iliul,'  ,vi,l  hilitiul,-  in/.'  ,//,i/ixri  oj  11^/1/  ,i.wtisi,>ii  and  ./ir/imttion . 

.hxiiiii<ii/s,   J)  '.»  lon^^ittiih-  III!,/  hililiitlc. 

l-'ii|;\ll  I.  I.:      rW/     .  ,1,-.   f-(r'.")il,'   (   (  (.  "liVf; 


(".") 


2>'i)  long. 


275 
38u 

285 

21)1) 
3<J5 

3"" 
3"5 
310 

315 
320 
325 

33i> 
335 
34') 

345 
35'> 
355 

o 
5 


'5 
80 

25 

3" 
35 
40 

45 
5" 
55 


+  •>33  -t- 
■  I3> 

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